It is hard to engage in "what if" questions, but the stability of the monarchy had been on shaky foundation since the end of the first World War. Certainly an argument can be made that it was Victor Emmanuel's tacit support of fascism that doomed the monarchy, but forces had been in play for years before then.

Italy was not in a shiny, happy place as a result of the outcome of the First World War - and this is the first area in which you see cracks in the ability of the Monarchy to stick. The War had been sold as a means to bring national honor, and instead had nearly broken the state. I'm being careful to say 'state' and not just 'nation' because Italy was soon to become a hotbed for potential revolution. While Italy eventually was on the 'winning' side, it is incredibly difficult to say that Italy won. In fact, an argument can be made Italy only won on a technicality. Desertion within the military was high, and a sizable amount of the military deserted at points with the socialists pushing heavily for soldiers to actively seek defeat on the battlefield. They became known as disfattisti, but whom the right wing and eventually the Fascists would brand vigliacchi - cowards. The socialists were actively seeking to cause a revolution as seen in Russia by the war's end.

John Foot sums up what the so-called Great War meant to the nation quite well in his excellent work Italy's Divided Memory:

For the socialist and anarchist left, and also for many Catholics, the war had been a “useless slaughter,” as Pope Benedict XV called it in 1917. For the nationalists, many of whom were to become fascists, the conflict had been heroic and glorious. These two visions of the past, and of the world, clashed in every corner of the country.

It is within this vacuum that the Fascists are able to rise to power by seizing upon the other sentiment in Italian society: that it was the socialists, and the disfattisti who were to blame for the disastrous defeats at Carporetto. The left, meanwhile, proudly wore this as a badge of honor, and used it as an example of the sign of a revolutionary class struggle. Unfortunately, the left's notion was a political miscalculation as the Fascists were able to recast the memory of the war as a clash between those who had run away, and those who had fought. Among many reasons, this was how Mussolini was able to spin his way towards power. Most tellingly how important this struggle between the socialist left and nationalist right was, were Mussolini’s first words on taking power in October 1922 to Victor Emmanuel III were “I bring you the Italy of Vittorio Veneto” - the site of Italy's defeat of Austria-Hungary that in effect ended the Austro-Hungarian Empire.

With Italy's splinter in 1943, these memories rang up again. Combined with the monarchy's support of Mussolini, it was almost an inevitable conclusion that there time had run it's course.

There were no exit polls asking why people voted to abolish the monarchy in 1946. However, there have been attempts to interpret the reasons behind the vote.

What emerged from the results was a strongly monarchist south and a republican north. It is generally accepted that this was a direct consequence of the German-occupied pupped state propped up in the north after the King's armistice following the allied invasion in 1943. With the allied advance meeting the Germans halfway through the peninsula, the south never had to live through a foreign occupation. The King's decision to leave Rome and seek refuge behind the allied lines in Southern Italy left many in north feeling angry and abandoned.

Then again, there is fundamentally nothing that the King could have done in 1943 to avoid a military intervention by Germany. But although the more immediate causes were the hardships brought about by military occupation, there is of course an argument to be made that had the King showed a stronger spine against Mussolini in the 1920s, maybe there wouldn't have been a vote in the first place. Your question does have very many variables.