The resolution at the group’s annual meeting in Belfast was passed by 364 votes to 163.
It had been proposed by the body’s Greater Glasgow group and mandated the organisation to lobby the Government on the issue.
The RCN remained neutral in 2016 when the first Brexit referendum was passed and one member at Tuesday’s meeting argued it should have remained so. But the vote makes it the first major trade union to support a second referendum.
MPs will get a “meaningful vote” in the House of Commons on Prime Minister Theresa May’s terms negotiated for leaving the EU.
Neither the Conservatives nor Labour back carrying out another poll.
The UK is due to leave the EU in March 2019, irrespective of whether MPs back or reject the terms of the deal negotiated by the Government.
The RCN debate was held in Northern Ireland, which shares a land border with the Republic of Ireland.