After the killing of Sollozzo, Vito stays in the hospital for some more months while the Mafia war is going on, just like everybody expected. Now the combined power of four families against the Corleones was definitely more dangerous than Sollozzo, the Tattaglias plus a police captain. With Sollozzo alive Barzini wouldn't get his hands dirty. Cuneo and Stracchi are out of the game.
So by provoking a Mafia war, Vito's sons put his life in a greater danger than it was with Sollozzo. Why?

Why?

MICHAEL: We can't wait. I don't care what Sollozzo says about a deal, he's gonna kill Pop, that's it. That's the key for him. Gotta get Sollozzo.

CLEMENZA: Mike is right...

Either way, the war was inevitable. Since all the other families-as proved at the later meeting-had made deals with Sollozzo and wanted the drug dealing thing, the Corleones had no other solution, than to take quick advantage by starting the war.
Negotiation couldn't be a solution since the others had already made an attempt against Vito.
Sonny was right, Tom was wrong, and that's why he never became Michael's consigliere

After Michael's visit to the hospital, licensed, armed private eyes guarded Vito. Plus, after Mac was shot with Sol, the Police Commissioner probably thought it was politically advisable to have cops on hand to protect Vito, the better to prevent any further attempts on Vito that could prove embarrassing to the NYPD.

I'm guessing that, since Vito was totally incapacitated, and guarded round the clock, the other families would leave him alone and target Sonny, who was running the Corleone end of the war and was far more dangerous to them at that point.

"Gotta get Sollozzo"

If the Corleones had launched info about Mac to the papers and Vito was being guarded by private detectives and possibly (as TB is pointing out additionally by policemen), how could Sollozzo get Vito?
If the war was inevitable, the Corleones had the small advantage of having killed Sollozzo and Bruno (both of which played no role in the muscle structure of the families). But this advantage had been canceled out by losing the smartest guy in the family - Michael, who had to flee to Sicily.

I agree that after the information about McCluskey came out, the police made sure that Vito had the protection he needed. Remember during the war, the police were cracking down on all kinds of "business" operations to which they usually turned a blind eye, and that was hurting everyone's business. Barzini and the others, as Sol said. missed their chance. Besides, everyone knew how infirm Vito was, and that Sonny was more likely to do something stupid to lose the war.

This thread raises another question. How important was Solozzo in the scheme of things? I think Barzini was leading the charge to get the families into the drug business, and Solozzo happened to be in the right place at the right time. N.B. when Vito caved after Sonny's death they made a deal on drug distribution, and Sol was long gone.

Sol turns out to be a stalkiing horse for Barzini.

I don't find anything in the novel or film to indicate that the NYPD provided protection for Vito after Michael's sojourn to the hospital.

However, there is language in the novel and film explaining the Corleone reason to murder Sollozzo.

There is nothing to show the police protected Vito after Michael's sojourn to the hospital, but by implication after McCluskey was linked to the drug rackets, my guess is the police no longer impeded efforts to protect him.

Lack of impedence? That's quite a leap.

My point is that it was bad reasoning, because with or without Sollozzo they'd try to murder Vito.

No, I think they knew Vito was no longer capable of being a full time Don. He was no longer wirth the expense and bloodshed. Everyone knew Sonny was running things, and they also knew Sonny was a bad don. Even after Sonnny died and Michael tried to buy Moe Green out, Moe was going on about how Vito was sick, and how the Corleones were being moved out of New York. The consensus was that from the time Vito was shot going forward, the Corleones were perceived as weak.

I pretty much agree DT. However Danito, Vito's death was only pursued by Barzini and Tattaglia and only because Sollozzo offered the opportunity; there's nothing to indicate that the other families pursued it nor that Barzini and Tattaglia otherwise pursued it. As the script and novel clearly state, Michael and Sonny were convinced that Sollozzo was an exigent threat that required remedy.

It's interesting to read in the book that the plan to kill all the heads of the families was Sonny's idea.

As for the war provocation, I believe the Corleone Family really didn't have much of a choice in the matter given the assassination attempt on Don Vito's life. If Sonny had laid down to the other families the Coleone Family would have been crushed.

I agree the other families may not have been in on it as directly as the Barzini and Tattaglia families, but they have had to give their assent before the hit on Vito. For Sonny and Michael too, that was enough. For Hagen, who I usually defend, it probably would not have been enough in terms of launching a full scale hit on everyone.