The institution has spent £5.6 million developing a presence in Manhattan.
But, according to BBC reports, its application to the New York authorities for a licence to teach and award degrees has yet to be approved.
Labour said the campus was “a very expensive white elephant”.
University bosses insisted it would eventually repay the investment.
University chancellor Mohammad Yunus launched GCU NY in September 2013 when a 15-year lease on premises was agreed.
In April that year, the university announced that it had applied to the New York State education department for a licence.
At that time, it said the application was “progressing well” and anticipated the whole process would take “around 18 months to complete”, making it possible for teaching to start in autumn 2014.
But courses have not gone ahead because the university is still waiting for a licence.
The university’s deputy vice-chancellor, Professor James Miller, told the BBC he was “confident” degree-awarding status would be granted by New York State.
“We’ve been told that as far as they’re concerned there are no particular issues with our application,” he said.
In a statement, the New York state education department said it had several pending requests and “there is no established timeline for the completion of the review and consideration of those applications”.
Labour’s public services spokeswoman Jackie Baillie said: “What we have here is a university’s ambition going far ahead of what they can deliver in reality.
“And what we’re left with is a very expensive white elephant.”
Prof Miller said Labour’s criticism was “grossly unfair”.
“It’s a good investment as far as we’re concerned. The business case was robust. We assessed the risks at the time and decided that it was a risk that was worth taking. We will generate that money back. I’ve got no doubt about that,” he said.
A spokeswoman for the university added: “It is highly disappointing that rather than directly engaging with us to discuss our GCU New York venture, the first route chosen by Jackie Baillie MSP has been to secure media coverage to flag her concerns.
“Moreover, we are surprised at her unsubstantiated and inflammatory comments and that she should seek to undermine the legitimate ambitions of the University, which remains committed to deliver for our students, staff and the communities we serve throughout Scotland and beyond.”