Egg yolks make ice cream richer, smoother, creamier, and softer. (They are definitely not just for the color.) The whites don't provide the nice textural benefits, so you definitely can't replace two yolks with one whole egg. You also don't need extra ingredients to get air into the ice cream; air is added to ice cream by churning it. So there's no need to include the egg whites at all.
Specifically, egg yolks contain fat, protein, and emulsifiers (lecithin). By keeping the custard emulsified, i.e. avoiding separation even on a small scale into water and fat, they prevent larger ice crystals from forming. Cooking the custard also thickens it, helping give the desired texture.
I think you could add whole eggs to ice cream, but the egg whites don't really provide anything beneficial, so you'd have to for example replace 3 egg yolks with 3 whole eggs. You'd also need to strain it after cooking to avoid bits of cooked whites. (This is a good idea anyway, just with the little bits of white that tend to come with the separated yolks, but with whole eggs it'd be way worse.) And perhaps worst, you'd get a much stronger eggy flavor than you do from the yolks alone.
Note that it's also possible to make ice cream without eggs at all. Ice cream made with egg yolks, milk, and cream is called French-style, while ice cream made with more cream and no eggs is called Philadelphia-style. The Philadelpha-style keeps it soft enough by increasing the cream to milk ratio, i.e. adding fat. It's not a direct substitution, though: French-style is always going to be richer and creamier.
So you can make something good but different by reducing (or omitting) egg yolks and increasing fat in the form of cream, but the yolks are absolutely necessary in order to obtain the desired results in the recipes that use them. You could also add pure emulsifiers (e.g. soy lecithin) to make a smoother ice cream without eggs, but I think that's getting a bit beyond the scope of the question.
And it's no accident that fat is added in the form of cream, not something like butter or margarine. If you add pure fat like that, you won't be able to get it fully emulsified, even if you do have egg yolks. You'll end up with little bits of hard frozen butter/margarine in your ice cream.