I don’t think so. Barring really wild speculation about an alternate plan for Frodo, it’s just not possible for the Ring to get to Mount Doom without Sauron being distracted.
It’s very difficult to assume what would happen, but, let’s say Gandalf dies in Moria and does not return. When Merry is in Isengard, he grabs the Palantir. Nobody is there to stop him and take it away. In that case, Merry could, even probably would, have given a lot more information to Sauron about Frodo and the Ring.
Gandalf is not there to heal Theoden, true, but moreso he is not there to meet with Aragorn / Elrond / Gimli. They’ve tracked the hobbits into the Entwood. Where would they go from that point? They only go to Rohan because of Gandalf. Without going to Rohan, would Aragon have sought out the Paths of the Dead? Or, would he have gone directly to Gondor? In the latter case, he would have had extreme difficulty managing with Denethor (who’d likely have him thrown out of the city). Aragorn was only set in motion by Gandalf.
In any case, we can easily write off Rohan from the war effort entirely, and very likely no Armies of the Dead either. In which case, Gondor falls, Rohan falls, Sauron has essentially won the war. There are no great nations to oppose him at this point. The siege of Gondor would have been over by the time Frodo gets anywhere near Mt. Doom, yes? Now, Mordor would have been teeming with orcs and very likely the Ringwraiths at this point. Frodo’s task is nearly impossible.
But let’s just say for the sake of argument that the Ring is destroyed. This takes out Sauron and the Ringwraiths, but not the orcs, trolls, or evil men. These forces, even left leaderless, have some power structure without Sauron or his Nazguls; are they going to flee, or take over the temperate and productive lands of Rohan and Gondor? Gondor’s major cities are destroyed. Rohan in practice would be owned by Saruman, who would be the only major contender; he could very easily take over these forces and cause an awful lot of trouble. Saruman wouldn’t have been defeated, his armies would be occupying Rohan rather than warring against it, and would be intact.
In short it’s a pretty bad situation for the West overall. Definitely it could not be called a ‘win’ in any real sense; Gondor is essentially gone and Rohan is ruled by an evil wizard. Soon enough, the people of Rohan would resemble those of the East - manipulated by an evil power.
Gandalf was, all things considered, pretty much the keystone. You can imagine a win without Frodo, even without Aragorn could be arguable, but no Gandalf? I don’t think so.