Addressing the debate surrounding Islam following a summer of terror attacks and burkini bans, he stressed that French secularism was not at odds with the religion.
"Nothing in the idea of secularism is opposed to the practice of Islam in France, as long -- and that is the vital point -- as it complies with the law," Hollande said in Paris, stressing that secularism was "not a religion of the state that stands against all other religions."
"What we need to succeed in together is the creation of an Islam of France," Hollande said.
He said that this could be achieved through the new Foundation for Islam in France, a measure announced in the wake of the terror attacks to improve relations between the state and the country's large Muslim community, which accounts for between 7% and 9% of the population.
Longtime French politician Jean-Pierre Chevènement was appointed head of the foundation last month, CNN affiliate BFM TV reported.
Hollande said France also needed to create "a national association in order to obtain financing for the building of mosques and the training of imams."
"The republic cannot accept a situation where a majority of imams are trained abroad and sometimes don't speak our language," he said.
France's rules of secularism prohibit the use of state funds for places of worship, and there have been concerns about the radical vision of Islam practiced in some foreign-funded mosques. At least 20 Muslim places of worship have been closed due to extremism since December, French Interior Minister Bernard Cazeneuve said in July.
Hollande said that radical Islam had created "a fake state, led by real killers. It skews the Islamic religion to spread its hatred."
France is under a state of emergency introduced after the ISIS terror attacks in Paris
in November and extended following the Bastille Day terror attack
Hollande: Islam compatible with republican values
France has sought to clamp down on radical Islam following a string of ISIS-inspired attacks and concerns about the activities of hundreds of French jihadists who have joined the terror group's ranks in the Middle East.
The Interior Ministry says more than 2,100 French citizens are involved in jihad
, with 680 currently in Syria and Iraq and more than 200 having returned from the battlefield.
Hollande said the core republican principle of separation of church and state was conceived at a time when Muslims made up only a tiny proportion of the French population.
"The question is to know if the principles laid down more than a century ago remain relevant when Islam is the second religion of France," he said.
"The answer is yes, clearly yes. The vast majority of our Muslim compatriots demonstrate it every day."
President's swipe at right-wing rivals
In a possible reference to the furor surrounding burkini bans in a number of French towns
, enforced despite court rulings against them, Hollande said: "As long as I'm president, there will be no legislation of circumstance, be it inapplicable or unconstitutional."
He said the secular, democratic and pluralist values of the "French project" were admired by people throughout the world, and that the French should not lose their faith in those values nor sacrifice civil liberties in the face of the Islamist terror threat.
"Citizens of the world know that when you attack France, you are also attacking its liberty, its democracy, its culture and its way of life," he said.
"The danger would be that, faced with a challenge, France doubts itself, that it withdraws," he said. "How can one resist these attacks, build Europe, succeed in integration, if we do not believe in ourselves?
"It is the trust we have in ourselves and our values which will enable us to triumph over terrorism."
Without naming them, Hollande also took a shot at his right-wing political rivals -- who have attacked his track record on security and called for a harder line on Islamists -- saying that any suspension of civil liberties threatened the values of the French republic.
"We know that each time democracy doubts itself, populism, demagogy, nationalism is strengthened," Hollande said.
No declaration of re-election bid
Hollande said that France, which will hold presidential elections in 2017, faced crucial questions about its future.
"It's the fight of a lifetime," he said. "We are in France, a country whose choices will be decisive for the future of Europe, for its very existence."
France's war on terror has been painful, he said, acknowledging the country's Muslims had suffered due to the actions of fanatics, too.
"We have paid a heavy price. Two-hundred thirty-eight dead and many injured," he said.
But France would prevail, he said. "Democracy will always be stronger."
The speech was being watched closely for a confirmation that Hollande, who is faring dismally in opinion polls, intends to seek a second presidential term in the 2017 elections.
While he made no such declaration, the center-left politician vowed not to "let the image of France deteriorate during the next few months or the next few years," suggesting he may seek to remain in office.
Hollande's predecessor, the center-right politician Nicolas Sarkozy, announced his candidacy for 2017 last month