The proposed law, presented as the first major social reform of Francois Hollande's presidency, would grant homosexual couples the right to adopt children but not to use assisted procreation methods such as artificial insemination. France's parliament is due to vote on the proposals in the middle of next year.
The draft was a compromise, leaving out the complex issue of assisted procreation to ease its way through parliament. But left-wing deputies have vowed to amend the text to include it.
Leaders of all major faiths and some conservative deputies have vigorously denounced the plan and lay Catholic groups have announced street demonstrations against it next week.
"This is an important step towards equality of rights," family minister Dominique Bertinotti told reporters after the cabinet meeting adopted the draft to allow "marriage for all", as its supporters describe the reform.
A government spokesperson said Hollande told the cabinet the reform would be "progress not only for a few, but for the whole society", a clear response to a charge by Paris's Cardinal Andre Vingt-Trois that it was "a fraud" favouring a tiny minority.
The head of France's Roman Catholic Church told his fellow bishops in the pilgrimage town of Lourdes last Saturday that same-sex marriage would upset the equilibrium of French society and harm children growing up without a father and a mother.
Jean-Francois Cope, secretary-general of the conservative UMP party, said the government's plan would "play havoc" with the Civil Code, which would have to be re-worded to remove gender references from passages dealing with family issues.
Their criticism has dominated the public debate in recent weeks, prompting a slight dip in voter support to around 60% for homosexual marriage and around 50% for homosexual adoption.
If the law is passed, France, a traditionally Catholic society where churchgoers are now a minority single-digit percentage of the population, would become the 12th country in the world to allow homosexual marriage.
France legalised gender-neutral civil unions in 1999 and almost as many are contracted now as traditional marriages. But only 4% of those are among same-sex couples. In recent weeks, the Catholic Church and France's Jewish, Muslim, Orthodox Christian and Buddhist religious minorities have been especially severe in criticising the possible options of adoption by and assisted procreation for homosexual couples.
In separate statements they have avoided using religious arguments and based their criticism on what they said were social, psychological and legal problems.
The government originally underestimated opposition to the reform but decided to extend the time for parliamentary hearings in January when critics accused it of trying to stifle debate.
The pro-gay rights group Inter-LGBT was due to hold a rally outside the National Assembly on Wednesday evening to demand that assisted procreation, which is currently only available to married heterosexual couples, be included in the draft law. Some French lesbians who desire children now travel to neighbouring Belgium for artificial insemination.
Surrogate motherhood is illegal in France and the draft law would not change that, meaning that homosexual men would not be able to engage a surrogate mother abroad and have the child recognised as their own on return to France.
Adoption rights would help homosexual couples who already have children, because the partner with no biological link to the child could legally become a parent.
Estimates put the number of such cases at already between 40,000 and 200,000. But it is unlikely that many homosexual couples will be able to adopt children unrelated to them because there is a shortage of children for heterosexual couples already seeking to adopt.