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A number of people have attempted swimming in heavy medieval-style armor as experiments, with videos available on YouTube or similar sites. I've never seen one where the person actually succeeded. They tend to be at a point where they speculate that with more strength and training they could do better, but... no success so far.

See here for a sample of three such videos.

Fat warriors could carry more.

We will upfront dismiss assertions that fatness is incompatible with fitness which is of course ridiculous. Next: @Therac's answer above covers upwards thrust produced by arms and legs to counter additional weight.

Let us consider your fat warrior. We will use Andre the Giant as our example. Andre was a 201 kg warrior with 27% body fat. That is 54kg of fat. Fat is 90% of the density of (fresh) water and so 1 kg of fat + 100g extra mass will be neutrally buoyant.

In other words every kg of fat your warrior will, in addition to itself, float 100g of additional mass or 10% of its own mass. 10% of 54 kg is 5.4 kg and so by virtue of the extra fat Andre could float with extra weighty metal gear and compared to some lean marathon running warrior. This is separate from any effort expended kicking to stay afloat; Andre would just bob around.

You can scale that up as far as you want although I expect diminishing return for a standard human frame at very high amounts of fat weight. These doughty warriors will of course float even better in salt water than in fresh by virtue of the higher density of salt water, but their gear would get rusty faster.

If your soldier knows he will be swimming in armour, you can provide him with an air-filled float. The float can even be made of steel! This can either be a permanent part of his armour, or an attachment.

Some suits of armour already have a bulging breastplate which could house a float (though this may not be the best buoyancy distribution as it could cause him to flip over on his back.

A litre of steel weighs 7.5kg, so for every litre of steel you would need 7.5 litres of air (displacing 7.5kg of water) for neutral buoyancy.

As you add bulk you add more frictional resistance, but this can be compensated by flippers to improve swimming efficiency. And a shield that doubles as a body board / surfboard - why not?

Other answers here seem to looking at the ability to swim under load from a diving perspective, I believe this may not be applicable as many diving techniques rely on flippers.

Water polo players and synchronized swimmers will be familiar with what is called the eggbeater kick, this kick is very effective at providing constant upwards force with reasonable energy demands. This technique is what allows water polo players to launch up out of the water to pass and take shots and for synchronized swimmers to lift their partners out of the water. From a personal perspective this is also what allows for lifeguards to lift and swim with a 9kg (20lb) brick used in training.

This paper suggests that the buoyant force produced by the eggbeater kick may range from between 10-20% of the person's body weight. This would mean that a warrior weighing 180lbs could potentially support themselves in the water carrying 36lbs of amour.

Note: eggbeater kick is primarily effective at providing upwards force and not as much horizontal force, so while a warrior could use this to support themselves in the water they wouldn't be moving quickly. They would also be at much higher risk in choppy water as their head height under such a load would be very close to the surface of the water and waves would cause frequent interruptions to their breathing.

I will go with a 0th order approximation: the human staying still in water, just floating.

The average density of the human body is $985 mathrmkg/m^3$, and the typical density of seawater is about $1020 mathrmkg/m^3$. (a) The average density of the human body, after maximum inhalation of air, changes to $945 mathrmkg/m^3$. (source)

This means that, for a $80 mathrmkg$ human, his body will occupy after maximum inhalation $0.085 mathrmm^3$ and will float displacing $0.078 mathrmm^3$ of seawater. This leaves an additional weight of $(0.085 mathrmm^3-0.078 mathrmm^3)times 1020 mathrmkg/m^3=7.14 mathrmkg$, which can be added without sinking, if the density of the added material is higher than the density of water.

Considering that one’s nose will sink underwater before the head, and that one cannot be constantly at maximum inhalation, the additional weight is slightly less.

I know you asked for steel armor, but Samurai had a special swimming style just for swimming in armor.

http://www.daitoryu.ca/html/kandan/012808_2.htm

Nihon Eiho, or samurai swimming, began as a fundamental martial
discipline, a military skill as highly regarded as horsemanship and
archery, and just as practical. As far back as the Heian period
(794-1191), swimming was a part of the formal training for the
Japanese warrior, and there is no lack of military stories involving
swimming skills.

In the Edo Period (1603-1867), the art of Eiho achieved its peak.
Several different ryu were already established, and teachers achieved
a recognized status.

https://pop-japan.com/culture/swimming-the-way-of-the-samurai/

During the time of the Sengoku, also known as the Warring States
period, samurai clans would often battle each other over rivers and
seas. In the 15th century until the 17th century, warriors would swim
across rivers while in their armor and helmet. In the 17th and 19th
century, the samurai would pass messages to each other by navigating
through the rivers, seas, and lakes of Japan. With a samurai suit
weighing as much as 44 pounds, the warriors were expected to know how
to fight on water, or at least to stay afloat.

There are even videos about it:
https://www.youtube.com/watch?v=WwDvJeP4WOg&feature=youtu.be

According to wikipedia for richer Samurai the breastplate and helmet where made from metal, so maybe they are comparable to European armors.

TL/DR: It is a skill you can learn.

If this was a planned swim with time to make something special, replacing much of the leather undeneath (which is an important part of the armour but has at best neutral buoyancy when wet) with a light wood would make things much better. Perhaps a composite formed of balsa on the inside to provide buoyancy and padding, with laminated alder, spruce or pine (less buoyant but stronger) against the metal. The idea would be to form plywood in place; 2D bends are easy but 3D bends are possible. You would of course need a non-soluble glue.

If the wood has an average density of 200 kg/m³ (i.e. mainly balsa) and is used to support 1.5 mm steel with a density of 8000 kg/m³ you'd need 13 mm of wood (1/2") to acheive neutral buoyancy. This seems a little large but not impossible at least for greaves and a cuirass; a stripped down version of the latter (steel breast-plate plus protection for neck and shoulders, with a thick collar of buoyant material to support the head). Strapping a helm and weapons to a shield a towing them would be a good idea.

Bamboo has also been used in the construction of armour (even occasionally on its own) and floats well; armour made almost completely of bamboo would easily float. If the swim was for a sneak attack from a lightly armoured force that would be one approach.

Even if you have netutral buoyancy or better, swimming encumbered is tiring and not quick. I've been known to swim in full kayaking gear; and even a buoyant, open helmet drags on the water, hence avoiding swimming in a heavy helm. A buoyancy aid is thicker than I've suggested for the armour but drags a lot.

There's a few things to bear in mind for this.

The first is that metal armour is expensive, so it's debatable how many soldiers would actually be wearing it. Clicking around indicates that plate mail would be the equivalent of between one to four years income for a man-at-arms (depending on the period in question - things like the Black Death had a huge impact on labour costs).

https://armstreet.com/news/the-cost-of-plate-armor-in-modern-money

And it's heavy both in and out of water - there's a reason knights tended to charge around on horses (above and beyond the fact that they're relatively defenceless when on the ground; there's plenty of tales of dehorsed knights being hamstrung and killed via a knife in the eye by pages and the like)!

Plus, as other people have noted, there's normally a lot of water-absorbent padding under the armour, which would further increase weight and make it harder to perform swimming motions; the extra weight would also stick around after they'd come out of the water.

And while body-fat might help with buoyancy, it's unlikely that many soldiers would be "bulky" enough for it to help. Life was a lot harder, food was scarcer and people were far more likely to have physical defects - for instance, British longbowmen often had skeletal deformities from their training. Plus, you know, army rations are rarely lauded by soldiers :)

So if there's a soldier in metal armour crossing water on foot, it's probably the result of a rout and they've probably been dehorsed and will already be suffering the mental and physical effects of defeat, exhaustion and injury.

Overall, I wouldn't give too much for their chances of making it to the other side...

On the other hand, it's interesting to look at the Roman army, with their standardised armour. A glance indicates that their mail armour (lorica hamata) weighed around 11kg (~24lb). They were pretty successful when it came to marching and conquering, so this may represent the best weight-compromise for marching/fighting/fording/etc.