Why didn't Frank avenge Omar?
"Obvious spoilers"
It goes without saying that Frank didn't take the death of his right hand lightly! As evident in the scene that immediately succeeded it. "You expect me to believe Omar was a stoolie. Because Sosa said so!?"
My natural assumption is that Frank realized Sosa was too powerful and starting a war would bring about an unnecessary amount of collateral damage. Even said as much to Tony! Not to mention that Frank is the type of gangster who would rather not make enemies (unlike Tony).
And correct me if I'm wrong! But according to "mob rules". You're not supposed to kill a made guy from another crew, unless you ask permission and the other bosses have to agree that the reason was valid. This is what happened in Goodfellas. Billy Batt's crew approached Paulie to inform him that Joe Pesci killed their guy, and under the circumstances had no choice but to "okay" the hit. I'm aware that Joe Pesci wasn't a made guy, but the principle's essentially the same.
Maybe Sosa thought he could get away with killing Omar, without permission from Frank. Because he couldn't do anything about it anyway! Though it obviously spelled the end of their professional relationship.
I can only imagine what Frank's men asked after they found out what happened. "I take it the deal's off!?" π