My character seems to always be gasping for air when I resurface but I have never taken any damage. Is there a certain amount of time that you can stay underwater without taking any damage?
TL;DR: Yes, you can die of suffocation from staying underwater too long. Waterbreathing equipment/potions can help avoid this, and Argonians are immune. All races will gasp for air when they emerge from the water - even Argonians. Stamina does not affect how long you can safely stay underwater.
Without special equipment, all races except for Argonians will begin to suffocate if they stay underwater for too long. Suffocation leads to a gradual loss of Health, which can eventually lead to death.
The only way to avoid this is to play as an Argonian, or wear equipment enchanted with Waterbreathing. Suffocation can be delayed by taking a Waterbreathing potion immediately prior to entering the water.
Regardless of race, or time spent under, your character will still gasp for air when they come out of the water. I've tested this myself with Altmer, Imperial, Nord, and even Argonian characters. This continues to occur even if you are using Waterbreathing equipment or potions.
While it would make sense, the amount of time you can hold your breath does not appear to be tied to your Stamina. This can be presumed due to the fact that your Stamina bar is not depleted while you are underwater. However, I did test this to be sure.
To test whether Stamina affects how long you can hold your breath, I started with a Level 81 Imperial who had a base Stamina of 350. I took her underwater, and timed the duration with a Stopwatch app on my phone. It took about 20 seconds for her to begin suffocating.
I then created a full set of Fortify Stamina equipment - Armor, Boots, Ring, and Necklace, all with +62 to maximum Stamina. This gave her a new maximum of 598 Stamina - a difference of 248, or a little over 58.5%.
I had her put on the new gear, verified the effects were applied, and took another dunk. She still began suffocating at around 20 seconds.
Yes, you will eventually take damage if you stay underwater for too long, unless you have Water Breathing. User testing seems to indicate its around 20 seconds or so before you start to drown, although I don't have concrete info on what effects this.
Argonians have Water Breathing as a racial trait, or you can get it via enchanting, alchemy, or by casting the Water Breathing spell from the school of Alteration.
You can drown in Skyrim, and all races (other than Argonians, who are amphibious) will start to lose health after about 20 seconds beneath the surface.
On the other hand, your character will always make the same, gasping, sound when breaching the surface of water, no matter how long you've been under, what race you are, or even if you're using waterbreathing magic.