More than 1,400 petition letters have been given to health managers in south Devon in protest at changes to local minor injuries units' opening times.
Teignbridge Primary Care Trust wants to cut back 24-hour cover at units in Dawlish, Bovey Tracey and Ashburton to between 8am and 8pm.
Teignbridge Council said local people feared the worst about the changes.
The Trust said it planned to increase the hours of its Newton Abbot unit and had invested £40,000 in services.
Teignbridge Council Leader Alan Connett said: "We know that the Primary Care Trust has more weeks to go in its own consultation, but there's a real anxiety that what this may mean in the longer-term is a withering of the vine and that the hospitals may be at risk."
Teignbridge Primary Care Trust added that the money it had invested was also to ensure more GPs could deal with minor injuries.