If a minister's daughter is pregnant with her second child and does not have a husband, can he keep his position as a minister?
The qualifications of a preacher do not include qualifications on his family. See: Preachers. This man's daughter's behavior does indicate that there was a problem in raising his daughter. He would most likely not be qualified to be an elder or a deacon.
If the church decides that it cannot support a man because his family's behavior reflects poorly on the church where he teaches, then they can decline to pay him.