The match was due to be played in Bucharest on July 10 but has now been cancelled. All players and staff are now isolating, according to a statement from Scottish Rugby.
The SRU said it had "every intention" of fulfilling the fixture against Georgia on July 17.
Those players and staff who have not tested positive will embark on a graduated return to training once their isolation period ends.
Scottish Rugby chief executive Mark Dodson said: "Player welfare is paramount in every decision that we make as an organisation and unfortunately we've run out of time to allow our players to recover and train properly for the Romania fixture.
"I feel massively for both our group of players and management, as well the Romanian Rugby Federation, however our squad's health and well-being must come first.
"We still have every intention of travelling to face Georgia the following week, subject to further medical review and testing, and we'll hopefully be in a position where a large section of the squad will be able to return to a graduated training regime next week."
Scottish Rugby chief medical officer Dr James Robson said: “We unfortunately now have a situation where we have several Covid-19 positive cases within the playing group and management, and as such we’ve made the correct decision to call off the Romania fixture.
“Simply put, you cannot hurry players or management out of isolation, and with the health and wellbeing of the squad very much in mind, we wouldn’t be in a position where we could safely travel to Romania and fulfil an international fixture.”
Scotland interim head coach Mike Blair added: “As a camp - of both management and players - we’re obviously massively disappointed that we’ll be unable to travel and face Romania next weekend, however it is 100 per cent the correct decision, taking into account the health and wellbeing of this group.
“It’s clearly been a disappointing couple of days, however, since meeting as a squad over a fortnight ago, the determination and resilience shown by this group has been evident to see, and we’ll move forward with that Georgia fixture very much in focus.”