I believe Roth acted against Michael in self-defense. Roth and Vito had been partners, but only in the Prohibition-era booze business, as Roth pointedly reminded Michael several times. But Michael had been horning in on Roth's non-NY gambling empire even before Vito died. The first business deal we see him in after Sicily was his attempt to buy out Moe Green--and when that failed, he killed Moe, Roth's closest pal. By the beginning of GFII, he has moved his operation to Nevada, owns or controls three hotels, is about to move Klingman out of a hotel Roth owns with the Lakeville Road Boys, and has designs on Roth's Cuban empire. Small wonder Roth might feel threatened. That's why Roth pretended to be MIchael's foster-father, all the while plotting to kill him.
Richie the bartender ordinarily might not have ratted out Michael Corleone. But Michael was about to be killed in Cuba, according to Roth's plan; and if Richie didn't know it, the Rosatos probably did. And anyway, the Rosatos were here-and-now for Richie, a clear and present danger. Michael was far away. As a rock/hard place, Richie might have chosen "Michael Corleone says hello" to "The Rosatos did it."
I, too have wondered: what happened to the Corleones' vaunted intelligence with the NYC police? Hagen tells Michael, "Our people with the New York detectives said Frankie was half dead..." Nice work, Tom, except that you found this out after your client, Michael, had his ass hung out to dry at the committee hearing. A guess: Ever-clever Roth heard first that Pentangeli survived, then set committee counsel Questad, who "belongs to Roth," to telling the committee chairman that he and all the members could get famous by exploiting this opportunity to hang Michael. So the committee leaned on the FBI to get the NYC police to hush up Frankie's survival and turn him over to them, so they could trap Michael into perjuring himself/
Finally: Carmine Rosato was drawing the gun to use on the cop, not on Richie.