The budget airline halted flights between the capital and Stewart in New York state following the Civil Aviation Authority announcing a ban on Tuesday.
It has now suspended bookings on the route, which is due to be axed on 29 March.
The European Union Aviation Safety Agency followed suit, with the US issuing an emergency order on Wednesday to ground the planes there too.
President Donald Trump said “new information” had come to light as part of an investigation into the crash of an Ethiopian Airlines 737 Max on Sunday.
All 157 people on board the plane were killed, including nine Britons.
Norwegian hopes to replace the planes with 737-800 aircraft on the Edinburgh route, which were initially used when it was launched in 2017.
However, this has not come in time for flights on Wednesday, Friday, Saturday and Monday, which have been halted.
A Norwegian 737 Max is based at Edinburgh, whose grounding has also affected some flights to Oslo.
Norwegian said it would seek compensation from Boeing.
An airline spokesperson said on Thursday: “Following the requirement by the European aviation authorities to suspend operations of the Boeing 737 Max, we are temporarily withholding further sales of the route to ensure we can meet our customers’ expectations and focus on supporting our existing customers who are affected by the aircraft grounding.
“We’re working to reallocate our fleet options with other aircraft types and re-route passengers travelling between Scotland and the US.
“Customers are being provided with care and assistance in addition to SMS and web updates, with the options to rebook or receive a full refund free of charge if they no longer wish to travel.”
However, some passengers complained of not being kept informed.
Dawn Fuge, co-chair of Perth Traders Association, tweeted: “It would be nice if they would actually contact the people who were due to fly this weekend and get them on a flight next week instead. But nothing. Can’t even get through to @Fly_Norwegian.”
Dr Thurai Rahulan, a senior lecturer in aeronautics at the University of Salford, disputed the decision to ground the 737 Max fleet, despite its involvement in a previous fatal crash in Indonesia last October.
All 189 on board died.
Dr Rahulan said: “Based on the limited information gleaned so far from the two accidents, banning the 737 Max by a number of countries as a precautionary measure seems excessive at this stage.”
Ethiopian Airlines said the black box with the voice and data records from Sunday’s flight would be sent to Europe for analysis.