"What we can't do is pit artist against artist," he wrote.
AFTRA has already reached a tentative agreement with Hollywood studios. SAG wants AFTRA members to vote against the deal, saying its approval will handcuff SAG at the bargaining table.
Both unions' current contracts are set to expire Monday, leaving Hollywood on edge about a possible replay of the 100-day writers strike that ended in February.
Results of the AFTRA vote are expected July 8.
Tom Hanks, Alec Baldwin and others have joined hundreds of actors in signing an online petition urging actors to ratify the AFTRA pact.
Meanwhile, Jack Nicholson, Viggo Mortensen and Holly Hunter have endorsed a SAG ad calling for AFTRA to return to the negotiating table to get a better deal.
Clooney called the fight counterproductive.
"Because the one thing you can be sure of is that stories about Jack Nicholson vs. Tom Hanks only strengthens the negotiating power" of the studios, he said.
Clooney also called on higher-paid actors to chip in a greater share of union dues and for 10 A-listers - "people that the studio heads don't often say 'no' to," he suggested, listing only Nicholson and Hanks by name - to sit down with studio heads once a year to "adjust the pay for actors."
SAG deputy executive director Pamm Fair called Clooney's letter valuable input. AFTRA and the Alliance of Motion Picture and Television Producers declined comment.
