There is "no appetite" for a second vote on leaving the UK and the party's next election manifesto will oppose such a move, he added.
It came after Nicola Sturgeon hinted she could be open to forming a Coalition Government with Labour in the event of a snap UK election, prompting speculation that an independence referendum could be the price of SNP co-operation.
Ms Sturgeon wants to hold a second vote on leaving the UK and will make a statement on her plans in the coming weeks. But control over the constitution remains with Westminster which would have to agree to such a move in the form of a Section 30 order.
But Mr Leonard told BBC Politics Scotland that a Labour Government would not allow this.
"What we're saying is what we said in the manifesto at the 2017 election which is that there is no case and we would not support a second independence referendum when we've only just had one," he said.
Jeremy Corbyn ruled out a referendum during a recent visit to Scotland, Mr Leonard added.
"What we're saying is what we said in the manifesto at the 2017 election which is that there is no case and we would not support a second independence referendum when we've only just had one," the Scottish leader said.
"My view is that the manifesto that we stand on at the next general election will contain a commitment that was contained in the 2017 manifesto which is that we will not agree to a second independence referendum
"There's no appetite for it either - people want a Labour Government that will get on with the job of investing in public services, addressing inequality and tackling homelessness."
"A Labour Government standing on a manifesto saying we do not support an independence referendum - because we've just had one - would not agree to a second independence referendum emerging from either the Scottish Parliament or any other quarter.
"My message is clear there is no case for a second independence referendum."
Ms Sturgeon hinted at a deal with Labour in a weekend interview.
Asked if she would be willing to go into coalition with Labour at Westminster if the results of a snap election enabled this, she told Greek newspaper Ta Nea: "We will be ready to contest any general election if one is called, and we would also be ready to work with other parties to oppose the Conservatives' "hugely damaging Brexit proposals."
Maurice Golden, Scottish Conservative chief whip, said Ms Sturgeon is planning a coalition with Labour and will demand a second independence referendum "as her price."
He added: “It is also quite clear that Richard Leonard's role in all this will be to sit in the back seat and do what he's told.
“Mr Leonard isn't prepared to take the fight to Sturgeon. He hasn't got a plan to stop a second referendum."