Does arguing mean he cares?

It isn't a sign in and of itself. All it means is that he wants to see that you agree with his side of the argument. He cares enough that he stays so that you can agree, or that you can fight it out until the truth of the matter is revealed, but he doesn't necessarily care for you in 'that' way. Guys have many ways of showing that they care, and this might be one of them for some guys, but it's not by any means the most definitive. What other signs has he given you? That's where you find out - it takes more than one sign.

I think a guy might argue because he cares, but he might also not argue because he cares. It depends on his personality.

Ex 1 - My wife is mad about me leaving my shoes all over the place - I argue because I don't think its a big deal and I want to do it again.

Ex 2 - My wife is mad about me going out with my friends all the time - I don't argue because I think she is right and I want to do better

Bottom line. I think you need to understand each guys personality and the topic before you can make a decision. I would suggest looking at the pattern of arguing to determine how you think his feelings might have changed over time. At the same time. I would personally use meaningful conversation as a barometer to determining if someone cares, not heated conversations that occur because of accusations, wrong doing, or displeasure.