Probably not but there already has been some talk about the clubhouse being divided between hitters and pitchers. Padilla and Byrd's history adds to that.
But I don't really think Padilla is long for this . I can see him easily being DFA once Cook is ready.
Baseball players are usually pretty good at forgetting on-field differences once they get on the same team.
Even off-field differences are easily forgiven. Brain Daubach was eventually well accepted by Tim Wakefield (there were interviews to this effect at the time) even though Wakefield was a union rep and Daubach was a "scab" player in 1995. The same thing happened in other clubhouses.
Shoot, pretty much no one of the Red Sox of the 1980s got along with Wade Boggs, thanks largely to his colorful off-field activities. But they somehow pulled together and won three divisional pennants in 5 years during the Boggs era.
I agree about Padilla, though. I'll be mildly surprised if he's still around by the ASB -- and shocked if he's on this team by mid-August.