The question is, did the military depose him using the demonstrations as an excuse, or did they depose him because they sincerely thought he was turning himself into a dictator?
That's a good question, and I haven't found an answer for that so far. Fact is, Morsi was never too much of a democrat, and he had ties to clientele with a deep green shade to it. I find it likely that the army ousted Morsi to prevent Islamism from rising. The army and Islamist radicals were always mortal enemies. My guess is that the military will tolerate a democratic government if it keeps the i-word out of the constitution.
Ideally, popular pressure should either cause Morsi and the Brotherhood to lose the next regular election, or - at worst - force a new election sooner than scheduled.
The problem is, I don't think anybody knows where the public's sympathies lie. Morsi was elected by an ever-so narrow margin, and only because the other option was a Mubarak supporter. How many people really do support the Brotherhood? What does the Brotherhood stand for after the revolution? Are the people cheering for Morsi's deposing the majority? Fact is, Islamists in Egypt are getting more radical by the hour. Many of the want Morsi back, and the military coup does give them some sort of moral legitimacy.