Steel's density of ~7.75 times that of water means you can discount 13% of its weight to buoyancy, but that's about it.
I have lifted 18 kg of collected dropped weightbelts from the bottom on one occasion, which put me at -10 kg of surface buoyancy after my gear weight and wing buoyancy was accounted for, and -14 kg at the bottom for the initial ascent (due to wing compression). This was a major physical effort and I had to hand the belts to the boat ASAP, but I was able to stay sufficiently afloat to ask for the assist.
Generally an experienced technical diver wearing jet fins can sustain about 12 kg of upward thrust, with fins, during an emergency ascent with failed buoyancy devices. Peak static thrust has been measured at 15-19 kg for ~90 kg body weight professional divers. Producing upward force is not equivalent to swimming, as your lateral speed will be very low. It's just a struggle to get to the surface, and you could probably brave a very narrow stream like that.
This is not the average. The average sustained thrust with fins was measured at about 64-69 N, or just 7 kg. Note that this is total prolonged sustained thrust, while practical swimming requires comfortably reaching up for air, but it could be done over a prolonged swim (wide river).
A practice among good swimmers called "monkey diving" involves wearing no BCD (buoyancy compensators) and compensating for buoyancy changes with swimming thrust - so this can be considered a practical swimming weight. The buoyancy at the beginning of a monkey dive can be -3 kg. This takes some effort, but is easily manageable with fins.
Without fins, humans produce very limited static thrust. I can stay afloat and swim with the aforementioned -3 kg of buoyancy without fins, but it's exhausting and slows me down. I can carry more briefly, but -3 kg is as much as I'd be willing to risk carrying across more than 400 meters without the ability to ditch the weight, and -5 kg in a do or die situation (the difference is major: -3 is struggling to get ahead, -5 is struggling to get a breath at all). My weight and swimming fitness would be in the range for the kind of character you describe.
Your average medieval soldier was certainly not a skilled diver, or an skilled swimmer, nor did they have any fins at all. This limits their ability to overcome negative buoyancy to -1 kg for most, and maybe -3 to -6 kg for the best swimmers, with a fairly large body for the era. This number is for swims across calm waters; large lakes, very wide or fast rivers, open sea can be challenging as it is (for that reason, everything above and below is for fresh water).
A sleeveless mail vest weighs about 5 kg. Armor is useless without a weapon (another 1-2 kg), so there's no point in bringing it even if one could. It's well possible to make lighter armor, for instance a steel plate with coverage similar to a SAPI insert, and such plates were sometimes attached to mail or leather. But it's unlikely that someone would bother doing that just for swimming, when it's more practical to supply waxed leather bags for buoyancy instead, or make a raft on the spot.
So the short answer is: Without fins or any buoyancy device - a simple log will do - you can't count on swimming over a decent-sized river with any kind of commonplace medieval steel armor that would be useful on its own.
A good swimmer without fins would still be able to cross rivers with their weapon and their leather armor pieces. They could carry something like steel bracers, but such armor is of limited use without torso protection. A short sleeveless mail vest (not a likely item for a soldier) would be the most a good swimmer could bring.
A shield would be a good flotation aid, and early styles (lime wood with little metal) would be able to support about the shield's own weight in steel. As such aids interfere with swimming, their buoyancy replaces dynamic swimming thrust rather than add to it, but it's a much more practical way of crossing rivers than rushing it.
If we go outside the military, professional swimmers such as pearl divers did exist at the time, and would be more capable. But finding one that turned soldier would be a one in a million occurrence and wouldn't make an army.
To put an upper bound on what's possible, modern Olympic-level athletes produce about the same static thrust without fins as a skilled diver with fins, so that ability to carry 10-12 kg across a river (only just, at risk to their life) could also be expected of them. This level of swimming fitness takes years of training and only came to exist with the reestablishment of full-time professional athletics in the early 20th century.