Would the petitioner be able to file their divorce in his home state?

If the respondent never lived in the state in which the petitioner now lives, that state does not have jurisdiction over him. Nor would it have jurisdiction over property in another state. The fact that it's a same sex marriage is irrelevant. A divorce is a divorce. Jurisdiction would be in the state in which the parties lived together or had property or in which the other party is willing to subject himself to jurisdiction if no real property is involved.