"Some who are delusionary want to pounce on the January revolution and think that they can undermine the stability that is growing daily or undermine the resolve that people have clearly forged with their will," said Morsi.
"We will deal with them decisively and there will never be a place for them among us," he told his supporters.
Morsi's government is widely thought to have failed to tackle any of the seemingly endless problems facing the country, from power cuts and surging crime to unemployment, steep price rises and fuel shortages. The declared aim of the June 30 protests is to force Morsi out and hold early presidential elections.
Morsi's allies say the protests have no legal basis and amount to a coup against his legitimate rule. They have been calling on opposition leaders to enter a national political dialogue to resolve the crisis, but the opposition has turned down the offer, claiming that previous rounds of dialogue did not yield results.
Spearheading the opposition to Morsi's rule now is a youth protest movement called Tamarod, or rebel, which claims to have collected millions of signatures of Egyptians who want Morsi to step down. Organizers say they aim to collect the signatures of more people than those who voted for Morsi in the June 2012 election.
Some of the hard-line clerics who support Morsi have branded Tamarod activists as infidels or heretics and sought to frame their movement as an act against Islam.