Hollande acted a day after the Constitutional Council threw out a legal challenge by the right-wing opposition, which had been the last obstacle to passing the bill into law.
The legislation also legalises gay adoption.
But while gay rights groups hailed the move, opponents of the measures have vowed to fight on.
Hollande made "marriage for all" a central plank of his presidential election campaign last year.
, he tried to turn the page on months of bitter opposition to the measures, arguing it was "time to respect the law and the Republic".
And he warned that he would tolerate no resistance.
"I will ensure that the law applies across the whole territory, in full, and I will not accept any disruption of these marriages," said the president.
French Justice Minister Christiane Taubira, who steered the legislation through parliament, has said the first gay marriages could be celebrated as early as June.
Marriages in France must take place in town halls, most of which take around four weeks to process marriage applications.
The issue of gay marriage and adoption has provoked months of acrimonious debate and hundreds of protests that have occasionally spilled over into violence and is unlikely to drop off the political agenda.
Although the Constitutional Council approved the bill , the International Day Against Homophobia, its opponents have vowed to fight on.
They have called a major protest rally scheduled for in Paris — and previous protests have drawn hundreds of thousands of people.
In April, the main right-wing opposition UMP party of former president Nicolas Sarkozy challenged the measures on constitutional grounds immediately after deputies passed the bill in parliament.
But Friday's statement by the Constitutional Council said same-sex marriage "did not run contrary to any constitutional principles," and that it did not infringe on "basic rights or liberties or national sovereignty".
Reacting to the ruling , UMP party chief Jean-Francois Cope told TF1 television: "It is a decision that I regret, but that I respect."
But late , between 200 and 300 protesters gathered in central Paris to denounce the ruling backing the bill and calling on Hollande to resign. One police officer was injured after a flammable liquid was thrown in his face.
Earlier, a group of bare-chested men wearing white masks staged their own protest against gay marriage on one of the bridges over the Seine. They call themselves the "Hommen" — a riposte to the bare-breasted feminist protesters known as the "Femmen".
Gay rights groups hailed the decision as a watershed.
"Now it's celebration time," said spokesman Nicolas Gougain of the LGBT association representing the lesbian, gay, bisexual and transgender communities.
But gay rights watchdog SOS Homophobie added: "Our country has taken a great step forward today although it's regrettable that it was taken in a climate of bad faith and homophobic violence."
The issue of gay marriage has divided France, which is officially secular but overwhelmingly Catholic. Protests against the bill drew hundreds of thousands, with a handful of hard-core protesters clashing with police.
Last year, the proposals seemed to enjoy solid majority backing among French voters.
But as the opposition campaign got into gear, more recent polls indicated a shift of opinion to the extent that the electorate is now fairly evenly split on both gay marriage and adoption.