In the Agamai Arc, Aizen's Captain initiation/Creation of the vizard Flash Back and during the invasion of the soul society other rules were mentioned, The rules are: -One must achieve Bankai (It was mentioned that Squad 11 rules are different and that killing the current captain voids this rule, seems like the rule was just placed there to ensure that any new captains werent just the strongest of what could be a class of weaklings but to ensure that they were of a certain rank in terms of Reatsu ) -One May Defeat a captain or kill them (Not sure if this one still applies but it only applies to Squad 11 and it is the same way Kenpachi became a captain.) -One must win a duel against another Captain before an audience of X amount of captains -One must be voted to be a captain by majority rule by the current captains. So far ichigo has defeated Kenpachi and Byakuya two captains famous for being undefeatable. and he is probably favored by all of the captains, maybe except for Soi Fong, and Mayuri then agin who isnt. Also he has achieved bankai and he has be noted countless times by captains and even Gen-Ryu himself to be of quiet remarkable strength. He technically defeated kenpachi which makes him eligible to be captain of squad 11, probably would win votes from most of the captains. So by those rules which all appeared in anime and manga, Yes Ichigo qualified.
Edit by Hajike Tobiume: The requirements for becoming a captain have been revealed. The first way is to take the captain proficiency test, which requires the demonstration of Bankai to at least three captains, one of which must be the captain-commander. This is the most common way to be promoted to captain level. The second way is to get a personal recommendation from at least six captains, and approval from at least three of the other captains. The final way is to defeat a current captain with at least 200 of that division's members to bear witness to the fight. Even though Ichigo has defeated captains in the past, it has not been in front of at least 200 of that captain's men. However, he has achieved bankai, and therefore would probably pass the captain proficiency test. I think it's possible that some captains might recommend him to be a captain at some point, however he's quite young and inexperienced, so it isn't likely to happen. So if he were to take the captain proficiency test, he would probably be able to become a captain, but I'm not entirely sure that he would be allowed to take it, seeing as he is not part of the Gotei 13. Not to mention he probably wouldn't want to leave his friends and family, as he's just not that kind of person.
He probably won't be allowed to become a because he is seen as a vizard by most people. Most people who are vizards were former captains or lieutenants but were banned from Soul Society because they gained hollow powers.
Well, hey, that was when Central 46 was in charge. I'm sure that at the end of all this, if Ichigo was willing to learn Kido and such, Captain-Commander Yamamoto would be willing to make him a captain. Seireitei has three captain positions to fill after all. Ichigo's somewhat uncontrollable spiritual energy shouldn't be a problem either. Kenpachi's a captain after all. The only things that are really not in Ichigo's favor are his young age and lack of experience in Soul Society. But Hitsugaya was really young when he became a captain. He even admits that his power isn't as mature as the other captains'. In the end the only real problem is that Ichigo is still a living human teenager in the Human World.