There are several sides of this question to take into consideration. The first is that while the Pope was a divine sovereign, his actual powers could be very limited without allies. The other cardinals were all powerful men born from powerful families, and while at this time most cardinals were Italians (about 3/4 of them at the time of the 1503 conclave that elected Jules II) we need to remember that Italy was divided into realms, republics and duchies with each their own governments and that the Italian cardinals often came from the families of the people ruling those realms, republics and duchies. The politics were ever-changing and often precarious; many of these men hated each other or were allies in name only. The Pope did not gain allies by being his holy self but by giving favours. Starting a war would have meant making many enemies, not only in the land that he attacked, but right beside him in the College of Cardinals.

The implication of the Papal Army in a war had to be justified to be accepted. During the reigns of Alexander VI and Jules II, that justification was the Papal States themselves. A big part, the Romagna, had all but declared its independence from Rome. Officially, the conquest of Romagna was about making it go back to the Papal States. Outside of the people directly involved (the Romagna and Venice), there was no reason for the rest of Italy to go against the Pope on that front. But if the Pope had tried to conquer a territory that didn't already belong to him, like Naples, that would have been another issue entirely.

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Of course, that's the simplified version, as the involvement of the French King Louis XII in the Italian wars did have effects on the Papal States's territory expansion. Sarah Bradford writes in Cesare Borgia, His Life and Times p. 148:

(Louis claimed that Florence owed him money for the war against Pisa in June-July 1500.) Indeed the Borgias guessed that Louis might not be averse to using the threat of il Valentino [Cesare Borgia]'s army to frighten Florence into paying up, although they also knew that he would never allow a direct attack on the city itself.

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So the first answer to your question is that waging wars left and right would not have been a good idea for the Pope. They would not have been legitimate and he would have made too many enemies. The Papal Army was not strong enough to vanquish them all.

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On the subject of excommunication, however, that's actually something Jules II did. When he went against Romagna in 1508 he also tried to expand the territory by taking land from Venice. The Republic wasn't happy with that, obviously, and Jules II excommunicated Venice in 1509. When the time came in 1510 to chase the French out of Italy, he made up with them and gave them back the cities of Faenza and Ravenna. In response, Louis XII tried to destitute the Pope with the 1511 Council of Pisa. And then Jules II excommunicated four cardinals involved in the council, two of them Spaniards. Which angered Spain. So we can see how this type of machinations can be endless. In essence, excommunication was indeed a weapon of the Pope, but during the Renaissance it had lost most of its might. It could be resolved quickly after the conflict between the two factions was over, and since there were always new conflicts emerging everywhere, it wasn't good for the Pope to stay enemies with his powerful neighbours for long. And then a few decades later, Henry VIII preferred to create his own Church to be released from the grip that the Pope had over him.

Edit: some typos