It's not clear to me that the Gallos wanted to take over the Profaci family, or that Gambino backed them.
Their initial beef with Profaci, ca. 1960, was that Profaci didn't give them territories he promised them in return for whacking a guy who wouldn't pay tribute to Profaci. They kidnapped several high-ranking members of the Profaci hierarchy, but released them after the Commission ordered them to do so--and ordered Profaci to give them the territory he promised. When he welshed again, the war started up. I don't know that Gambino took sides at that point.
Later (ca. 1963-64), after Profaci died, Joe Magliocco took over. He and Joe Bonanno conspired to whack Gambino and Tommy Lucchese. Joe Columbo, a Profaci captain, betrayed the plot to Gambino. The Commission ordered Magliocco to pay a fine and step down--in favor of Columbo, not the Gallos.