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Since Ronald Reagan, the Republican party has had three main pillars:

1. A low tax, somewhat libertarian group.


2. A pro-military group.


3. A moral majority, pro-life group.

The moral majority group is heavily based in rural areas where gun ownership is common.

The pro-military group includes many current and former service members who own guns.

The libertarians are historically pro-gun.

Until relatively recently, there were many pro-gun-rights, anti-gun-control Democrats. As the party has lost hold in rural areas, it has lost almost all of those (Joe Manchin is an exception).

The Democrats modernly are an urban party. In cities, guns are much less used. In fact, it is common for it to be illegal to fire a gun except in immediate self defense in most places in cities. The police are closer (so self defense is less of an issue). Fewer people in cities own guns legally.

Both the Republicans and the Democrats want to restrict things. The Republicans want to restrict lifestyle choices, like sexual behavior and drug use. The Democrats want to restrict financial decisions. In terms of guns, like in finances, Democrats are less concerned about freedom from government restriction and more concerned about freedom from violence (financially, Democrats are most concerned with freedom from being unable to finance essential needs). Freedom "froms" that concern Democrats:

1. Financial need.


2. Gun violence.


3. Environmental damage.


4. Prejudice (based on skin color, gender/sex, sexual orientation, etc.).


5. Unfair labor practices like being fired without due cause.

Government restrictions on abortion are the odd choice out from this perspective, but feminists (defined here as women worried about prejudice against women) feel strongly that the government should not have control over their bodies. Abortion is also often a backup plan for poor families that may not be able to afford quality birth control. Richer families are more able to handle unplanned pregnancies or travel somewhere where abortion is legal.

In the United States, gun control is a modern phenomenon. At the founding, the chief worry was that a tyrannical government would take away guns. Most adult men owned guns, even in comparatively urban areas. Organized militias sometimes required able-bodied men to own guns.

In the early twentieth century, Democrats began shifting from a rural, Southern party to one with many urban members, particularly in unions. In 1968, the traditional Southern members split off and supported a separate presidential candidate. In 1972, they supported the Republican Richard Nixon. By the end of 2014, almost all Southern Democrats had been replaced with Republicans.

Some other countries around the world:

Australia

The National Firearms Agreement was organized by the conservative/right-wing Liberal party.

Japan

In Japan, gun restrictions have been passed by the conservative/right-wing Liberal Democratic Party. Of course, gun restrictions have some history in Japan, an urban country where weapon ownership was long limited to the elite samurai.

New Zealand

The left-wing Labour party is currently promoting gun control (in 2019).

United Kingdom

Gun control was passed in 1903 by the right-wing Conservative party. And in 1997 by both the Conservative party and the left-wing Labour party (control of the government shifted during the year and each party added some gun control when it was in charge).

Communism

Most communist countries implemented gun control strictly. They were of course left-wing financially but were often right-wing on issues of crime and morality.

Trying to derive policy positions from first principle doesn't really work, especially not when they're so strongly driven by identity in the present.

But the key thing in gun control is to understand freedom from. Referring to FDR's speech on WW2 on the "four freedoms":

The first is freedom of speech and expression—everywhere in the world.
The second is freedom of every person to worship God in his own way—everywhere in the world.
The third is freedom from want—which, translated into world terms, means economic understandings which will secure to every nation a healthy peacetime life for its inhabitants—everywhere in the world.
The fourth is freedom from fear—which, translated into world terms, means a world-wide reduction of armaments to such a point and in such a thorough fashion that no nation will be in a position to commit an act of physical aggression against any neighbor—anywhere in the world.

The last of these is the important one. The rise of mass shootings has led to a desire for freedom from fear of being shot at, especially at school. The number of people affected by these tragedies is only increasing; there are even a few "second generation" school shooting survivors now.

Achieving this freedom can only come by a reduction in the availability and prevalence of guns.

However, the politics goes much further back, before the current mass shooting problem. Some people date it to the 1960s and the Kennedy assassination. Either way, the NRA became explicitly Republican. This included significant financial support to candidates. See this article from 1995! Surprisingly little has changed since then but the names.

Once the partisanship was entrenched, it became unshiftable.

The Republican Party has a conservative ideology which is characterized (among other things) by respect for American traditions, of which private gun ownership certainly is one. Since debates on gun laws are most often about making those laws more strict, it only seems natural that a conservative party supports less strict laws, thus keeping the status quo.

I suspect this may be reflective of the big divide in the US being urban vs rural. rural areas have more use for guns while urban areas tend to be more socially liberal for a variety of reasons. The difference between the parties reflects differences in their supporters.

pjc50's answer is correct, but needs more explanation. Like most positions of American political parties, it has to do with historical affiliations between groups, not philosophical principles. Specifically for the issue of gun control, firearms manufacturers and distributors were drawn to the Republican party because that party was pro-business.

It wasn't always this way. This is explored in the PBS Frontline documentary "Gunned Down: The Power of the NRA":

NARRATOR: But for the NRA, the gun wasn’t always a political issue. It had once represented something for hunters and sportsmen.

(...)

NARRATOR: Then the assassinations of the ‘60s— John F. Kennedy, Martin Luther King—

Sen. ROBERT F. KENNEDY (D-NY), Presidential Candidate: My thanks to all of you. And now it’s on to Chicago, and let’s win there.

NARRATOR: —and Robert F. Kennedy. Many American cities erupted into armed conflict. In response, Congress passed the first comprehensive gun control law in decades.

The Gun Control Act of 1968 had wide bipartisan support. According to ustrack.gov, House Democrats voted 157 in favor and 79 against; House Republicans voted 79 in favor and 39 against. In the Senate, 39 Democrats voted in favor and 13 against; Republicans voted 31 in favor and 4 against; and the Yea votes represented 91% of the country's population. Republicans actually voted for gun control in 1968 at higher rates than Democrats. The vast majority of those opposing gun control were in rural districts, at that time mostly held by Democrats.

Frontline continues:

WARREN CASSIDY, Former NRA Executive V.P.: NRA people said, “Wait a minute. We’ve got— we’ve got other things to worry about than teaching guys how to shoot or how to hunt, and so forth, or collect guns.” And that’s when— that was the transformative period.

NARRATOR: It formally happened in 1977 at the NRA Convention in Cincinnati. As they got down to business, there was a showdown, hunters and sportsmen versus gun rights activists.

CBS NEWS: [May 21, 1977] The National Rifle Association convention in Cincinnati went into overtime last night, a stormy all-night session. When it was over, some dissident members had taken control of the 400,000-member organization. What it means is even stricter support for the right to bear arms and against gun control.

JOHN AQUILINO: The core of NRA’s political support comes from a very conservative Republican group of people. They’re the ones who give the money. They’re the ones that pay the freight for all the political battles. And they’re very conservative.

But mainstream Republicans were still not on board. After an attempted assassination, President Reagan stayed out of gun politics. But his press secretary James Brady was severely disabled by the attack, and Brady pressed for more gun control legislation. The Brady Bill was introduced during the Bush administration, but it did not come to a vote then. It finally passed in 1993 and was signed by President Clinton.

Gun manufacturers and retailers were infuriated by the Brady Bill. They perceived it as a threat against their business. If only they could buy politicians opposed to gun control, they could stop this threat. They had the money to do it. They had control of the NRA, and could use that as their political vehicle. But which party should they back?

These were business owners, and the Republican party had a long tradition of being pro-business. Furthermore, it was cheaper to flip rural Democratic districts to Republican, than to flip suburban Republican districts to Democratic. So the decision was made to back pro-gun Republicans in the next year's election. They sent out mailers to the NRA membership...

TIM RUSSERT, Moderator, Meet the Press: [April 30, 1995] Aren’t you concerned when you say, “Nazi bucket helmets, government thugs, kicking down doors, killing, maiming people”— aren’t you inciting people? Aren’t you willing now to apologize for the tone of this letter?

WAYNE LaPIERRE: Those words are not far— in fact, they’re a pretty close description of what’s happening in the real world.

PAUL BARRETT: And in response to that, many mainstream Republicans, George H.W. Bush being the leading example, said, “This is not the NRA I’m a member of.”

NARRATOR: President Bush resigned his lifetime membership in the NRA.

But it worked. The 1994 election flipped 54 seats in the House, giving Republicans the majority for the first time in 40 years.

And it has simply polarized and escalated between the two parties since then.

Although as others have pointed out, there really is no simple answer, one IMHO important factor is a basic difference in philosophy. Very simplistically, Republicans want individuals to take care of themselves*, while Democrats want the government to take care of everyone. So we have Democrats generally supporting tax-funded medical care along with various other social programs, and likewise wanting the state to defend the individual against crimes &c. Republicans think individuals should bear the primary responsibility for such things.

*Though unfortunatly, they tend to object when that self-reliance leads folks to deviate from what we might call "traditional family values".