It's interesting you should ask. For some reason I've actually had a couple of people tell me that I have "a warrior spirit" about me. I'm not exactly sure what that means, but the comment has come usually when someone thinks I've demonstrated inner strength under adversity or dealt with a conflict well.
I tend to associate the warrior with Athena of the Greek pantheon. It's a battle ethic, but not the same ethic as that of Ares. Ares' ethic was very much that of the aggressive alpha male: attack and conquer as a way of boasting; but in the end, Ares was cowardly. Athena's ethic, by contrast, combined battle skill with wisdom and restraint.
I suppose it has to do with what do you fight for, why do you fight for it, and what kind of integrity do you show when you fight, whatever the "fight" may be. If my friends are right with their comments about my own "warrior spirit", then I'm certainly no physical warrior. Maybe words are my weapons, who knows.
Incidentally, I consider the term gender neutral, not exclusively male. I note it because I noticed a tendency in this thread for folks to use masculine pronouns when describing warriors. Remember the Amazons, y'all...
I'd think it a little odd if someone just randomly came up to me and offered that they're a warrior. I'd think they were bragging about their fighting prowess, and I frankly don't consider bragging or boasting to be part of an honorable warrior ethic. If it should come up in a different context, like a more philosophical discussion or talk about mythos and various ethical systems or ways of living, I probably wouldn't find it so odd.