It's difficult to imagine what might happen in that situation. Like MeyerLansky mentioned, Gotti could have been killed by members of his own family who were sick of him.
If that doesn't happen, I could understand Gotti taking on just the Luchese family. However, by that time Amuso and Casso were on the run or in prison so there wouldn't have really been anyone to go to war with.
If he found out it was the Genovese and Luchese families together, I don't think he would have been dumb enough to take them both on. I don't think the other two families would have joined up with him either.
You know a story I love from this time period? The Lucchese hierarchy were having a sit down with the Genovese hierarchy. Venero Mangano led Anthony Casso and Vic Amuso through a labyrinthine network of tunnels in order to meet up with Chin. They were offered alcohol and Casso declined, but Amuso accepted, and Casso looked down on Amuso for it.
Where did I read this? If it was the book Philip Carlo wrote about Gaspipe then you can probably discount it.
It is from that book. According to Casso, a drunken Amuso raised his voice against the Chin and banged on the table. Casso had to calm him down
The sitdown does seem to have actually happened though. It concerned Pete Savino owning Amuso and Casso money. Savino said he was ordered to pay them.
