I wouldn't say so.
First thing, you might be using "Italy" in the sense of the Italian continental mainland; but politically, Sicily was and is Italy, and up until the invasion of Sicily, Mussolini's popularity had been shaken but he still was in the saddle. It was the invasion of Sicily - and the bombings of Rome - that convinced the other Italian powers that Mussolini had to be ousted and peace be made.
That said, while the new government immediately opened negotiations with the Allies as soon as Mussolini was taken away ("left" seems to assume he did this of his own free will - he was secretly arrested), the new Italian government was never going to be able to stand as an independent neutral, let alone as a co-belligerent on the Allies' side, without Allied boots on the ground. That's because there were already a lot of German boots on the ground. Things might have been different with a different Italian military and political leadership, but with what was in place, the coat-turning was a dog's meal, and the Allied landings would have been necessary in any case.