"Pro-choice"

Pro-gun: I support the right of others to choose to own a gun, but do not necessarily support them myself.
Pro-gambling: I support the right of others to choose to gamble, but do not necessarily support it myself.

Pro-X means pro-right to X.

There is no right to choose; "choose" is a transitive verb. There is, however, a right to choose X. Just like there is no right to punch. You may punch a brick but not a person. There is no right to choose to rob a bank. There is a right to choose what clothes to wear.

"Live" is an intransitive verb.

By the pro-abortion logic, anti-gambling people are anti-choice. "Pro-choice" would mean supporting choice in general. 

Sorry, but those who are pro-abortion don't get to claim choice as exclusively refering to abortion.