It comes after NHS Dumfries and Galloway was forced to suspend admissions for a week.
It also had to downgrade the casualty unit to a minor injuries facility because of a shortage of doctors.
The health board said it was working to minimise any further disruption but councillor Willie Scobie said the situation was unacceptable.
He has arranged the meeting to take place on Saturday morning at the town's Coronation Day Centre.
"It is very serious," said Mr Scobie. "We have had the closure, all of last week, of the Garrick acute ward.
"Then this week, completely out of the blue, A&E is downgraded.
"This is appalling - the people of Stranraer have already got to travel 75 miles to Dumfries, unnecessarily, for appointments and that's a serious concern."
NHS Dumfries and Galloway said the hospital was open and fully staffed but that a challenge remained in securing medical cover on an ongoing basis.
"We continue with our endeavours to recruit to the area," it said.
"The board will continue to keep the public informed about any disruption to the services at the Galloway Community Hospital as a result of these challenges.
"We would like to emphasise that we are working hard to keep any temporary downgrade to a minimum.
"The board's priority is the safety of patients at all times."
A public meeting has been called amid concern about services at the Galloway Community Hospital in Stranraer.