Hollande, trying to turn the page on months of bitter opposition to the measures, said it was "time to respect the law and the Republic".
The Constitutional Council approved the bill on Friday, International Day Against Homophobia.
The French court ruling clears the way for France to become the 14th country to legalise same-sex marriage. French deputies approved the bill in parliament last month after a several days of often stormy debate.
The main right-wing opposition UMP party of former president Nicolas Sarkozy immediately challenged it on constitutional grounds.
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".
However, the council said, gay adoption did not automatically mean the "right to a child": the "interest of the child" would be the overriding factor in such cases, it ruled.
Reacting to the ruling Friday, UMP party chief Jean-Francois Cope told TF1 television: "It is a decision that I regret, but that I respect."
Hollande, as he announced his decision to sign the bill into law as early as Saturday, warned that he would brook 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," he said.
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.
Late on Friday, 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".
Opponents of the law plan another major protest rally in Paris on May 26.
Hollande had made "marriage for all" a central plank of his election campaign and the measures initially seemed to enjoy solid majority backing among French voters.
More recent polls however suggest the vigorous opposition campaign shifted opinion to the extent that the electorate is now fairly evenly split on both gay marriage and adoption.
As the French bill got the green light to become law, a report by the European Union's Agency for Fundamental Rights (FRA) highlighted the problems that gay, lesbian, bisexual and transgender people still face across Europe.
It said two-thirds of the community were still afraid to show their sexuality in public -- and a quarter had been victims of physical or verbal attacks.
"Fear, isolation and discrimination are everyday phenomena for the LGBT community in Europe," the agency's director, Morten Kjaerum, wrote.
The report came as protesters in Georgia broke up a gay rights demonstration in the capital Tblisi on Friday.
Thousands of ultra-conservative Orthodox supporters led by priests broke up the rally in the deeply religious ex-Soviet state, smashing through police cordons and forcing activists to flee.
Police had to evacuate scores of gay rights activists from the city centre, as crowds of chanting protestors charged after them, hurling stones at the vehicles and beating on the windows.
Among the countries that have already approved same-sex marriage are eight other European nations -- the Netherlands, Belgium, Spain, Norway, Sweden, Portugal, Iceland and Denmark.