Are you allowed to drink wine that non-Jews touched? Why?
Even if it is not cooked, a non-Jew merely touching the bottle is not enough to prohibit it.
I am not writing a source because it is an inferred answer: If one would look up the Halachos of Yayin Nesech/Stam Yanum in Yorah Deah he will see that what is forbidden is poured wine (and the wine that was in the neck of the bottle as it was being poured and went back into the bottle when it was set down). The concern would be that the Aku"m had in mind to pour it for idolatry. But touching is not even an issue.
According to Kabbalah, one should avoid drinking wine that was even looked upon by a non-Jew. (The Lubavitcher Rebbe would have the wine covered by farbrengens to avoid this issue).
There is also a letter from the Lubavitcher Rebbe where he says that wine owned by a non-shomer-shabbos Jew is forbidden, even if there is a seal within a seal.
If it is Mevushal cooked than yes.