The bill was debated for only a few hours on Tuesday and Senate President Pietro Grasso said the discussion would restart on February 23 at the earliest.
Italy is the only major Western country that has not yet recognised civil unions for gay or heterosexual couples, continuing to withhold from them legal protections such as inheritance rights.
Much of the opposition has focused on a provision that would allow unmarried couples to adopt their partner's children.
Critics say this would encourage surrogacy parenting, which is illegal in Italy.
Latest polls have suggested some 70 per cent of Italians support civil unions. However, only some 24 per cent think same-sex couples should be granted adoption rights.
Hundreds of thousands of Italians protested against the legislation last month in a Rome rally backed by the Church.
Showing how polarised the country is, there have also been large rallies in favour.