Can a non-custodial parent sue for joint custody?

I went through a divorce that was not contested where I was given full custody of our child. Recently my ex has stated that they wish to have the child on overnight visitation. The divorce stated "no less than three day visits", which my lawyer told me meant the child did not have to stay overnight. The reason overnights are not okay with me is that my ex is an alcoholic that has shown irrational and dangerous behavior patterns in the past. Can my ex take me back to court for overnight visitation or joint custody? Am I legally obligated to allow him overnight visitation? If he can take me back to court, is there anyway I can prove my ex is an alcoholic? Does that have any relevance in court decisions?