I didn't say that christianity didn't have any influence at all but the influence of ancient roman and greek is more significant than christian influence.I'm not talking only about the roots of democracy but also Roman civil law influenced much of modern western laws, also the republican model of Rome influenced french and american revolution. Even human rights and justice were influenced by ancient Rome, for example romans were the first to come up with the concept that everyone who's accused of a crime should have the right to be heard and defend himself in a court.
As you can see in the three example you mentioned ( prostitution, abortion, gay rights) , Christianity influenced the laws in a negative way rather than positive. Canadians had to fight against their christian heritage to reform abortion laws and gay rights. Even with the ongoing debate about reforming prostitution laws, you can see that evangelical groups are the ones lobbying the government against decriminalization of sex trade . Romans had more positive attitude towards sex as you can see in the ancient brothels discovered in Pompeii