Sam, here is an article about UNSG selection process. There is an informal rule that the SG not be a national from a permanent member of the security council. The security council nominates from a "middle power" which rotates on a geographical region basis. for example: africa, asia, eastern europe, latin america. after the security council nominates the general assembly elects.
while technically an american can serve, it is an informal rule that the sg be from a middle power. all members of the security council would have to agree to his/her nomination.
http://www.unsg.org/role.html