Many companies list "positive attitude" as a requirement for the candidates, including job positions that have nothing to do with customer service or management (for which jobs it could be reasonably justified).

However, some people have major depression or an anxiety disorder that makes it impossible to maintain a positive attitude due to the mere nature of the illness. Would it violate the Americans With Disabilities Act or any other law to discriminate against such a person for their failure to maintain a positive attitude, given that the disability doesn't prevent the employee from doing their job? (I have to emphasize this part, as many answers simply ignore it).

Positive attitude, from the top definition in Google: "In general, having a positive attitude means being optimistic about situations, interactions, and yourself."

A few things that I have to clarify due to common misconceptions:

• having major depression isn't same thing as "being grumpy", as some people here suggested. Both major depression and anxiety disorders are among the most severe mental disorders and often result in a disability to some extent.
• not having having positive attitude doesn't necessarily mean having negative attitude; there is the third case in this false dichotomy - maintaining the neutral point of view
• even when someone's attitude is negative (i.e. pessimistic), that doesn't mean that they tend to be quarrelsome or unable to get along well with their coworkers
• about 3-4% of people in the U.S. have had more than one major depressive episode in their lives, but with proper treatment the illness can be successfully managed in some 60-70% cases (don't think of curing it completely, though. usually it's more like treating type 1 diabetes with insulin, or the ability to move around using crutches for a person with limited mobility), or at least well enough to be employed.
• the above point proves that there are at least 5 million people in the U.S. who have major depression, but are able to get their job done without having "positive attitude".