Daniel Johnson said it is wrong for Labour to be in the position where its policy on Scottish independence is determined by party leaders south of the border.
It comes after Scottish Labour suffered another disastrous defeat in this month's general election - with only one MP elected.
While Labour in England lost seats to the Conservatives, north of the border they lost six to Nicola Sturgeon's SNP, with just Ian Murray elected as the MP for Edinburgh South.
Mr Johnson, the MSP for Edinburgh Southern, told The Edinburgh Briefing Podcast that similarities between the party north of the border and their English counterparts hit their electoral challenges at this month's election.
He said: "The Labour Party doesn't have a right to exist - and at the moment I think we're teetering on the edge of irrelevance.
"The first point of principle, and what I've been saying to colleagues, is that we simply cannot have a situation where in the biggest single issue facing Scottish politics, which is the independence question, that our position on that gets decided by the UK leadership.
"That cannot continue and that is my message to the leadership in the UK and it's my message to the leadership in Scotland."
The UK leadership of Labour has been accused of undermining the Scottish party, with Jeremy Corbyn and John McDonnell repeatedly refusing to rule out a fresh vote on Scotland's constitutional future.
Mr Corbyn repeatedly said he would not allow another vote "in the early years" of a Labour government during the election campaign.
Mr Johnson said a win for the SNP at the ballot box in 18 months could open the door to another plebiscite, despite his opposition to the idea.
He said: "We have a precedent and that is a majority at the Holyrood election. Those are only 18 months away.
"If those conditions are met, I will still disagree with independence, I will still say that I don't want another independence referendum, but if those conditions are met, then those of us who believe in the claim of right and devolution will have to acknowledge that there will be a mandate."
Mr Johnson also said the party was "sowing the seeds of doubt" on both Brexit and independence during the campaign, adding: "In Scotland I don't think we successfully articulated a differentiated message from the UK party - that hurt us."
Meanwhile, former Scottish justice secretary and newly elected SNP MP for East Lothian Kenny MacAskill said Labour must support "Scotland's right to choose".
Writing in the Scotsman newspaper, Mr MacAskill claimed the independence referendum in 2014 had "pushed many and most of the current leadership into an irredentist unionist camp".
But he added: "The suggestion that it's almost a sacred duty of Scottish Labour to sacrifice itself for the union is a total absurdity that besmirches the legacy of Hardie and others.
"Backing independence isn't required but supporting Scotland's right to choose is essential. Do that and they have a place, reject it and they're doomed."