Written by Edgar Rice Burroughs in 1912 and long before space flight, it was stated John Carter would be stronger on Mars than he was on Earth. If he stayed on Mars, would he remain strong, or would he lose strength as his muscles atrophied in the lower gravity?
From NASA: Your Body In Space: Use It Or Lose It
In order to stave off bone and muscle degeneration, astronauts engage in three primary exercises:
Cycle Ergometer: This is like a bicycle, and the main activity is pedaling. It is used to measure fitness in space because it's easy to check heart rate and how much work is being done.
Treadmill: Walking or jogging on the treadmill is like walking on Earth. Walking is the single most important way to keep bones and muscles healthy. Because the lack of gravity tends to make people float, harnesses are attached to the astronauts to hold them to the walking surface.
Resistance Exercise Device (RED): The RED looks like weight-lifting machines you may see on television. To use it, astronauts pull and twist stretchy rubber-band-like cords attached to pulleys. The RED can be used for a total body workout. From squats and bending exercises for the legs, to arm exercises and heel raises, astronauts can do them all on the RED.
So, basically cardio and some resistance training to keep muscles and joints moving.
While highly regarded and eventually being considered upper class on Barsoom/Mars, John Carter does not live a sedentary lifestyle by any means (at least in the first three books - I haven't read beyond them). If anything, his activities and battles on Barsoom are probably more strenuous than his experiences on Earth.
It should also be noted that the ISS and space shuttles are microgravity environments (essentially 0 gravity) whereas Mars has a gravity strength equivalent to 38% of Earth's, so the rate at which his body/muscles would deteriorate would be significantly slower than that of an astronaut. Thus, he wouldn't need to exercise as much as astronauts in order to maintain his current form either.
If Carter were to ever retire and live a quiet life in a palace and never fight again, then it is possible his body would deteriorate over time - but that hardly sounds like something he would (or could) ever do.
I imagine he would lose his extraordinary strength if he stayed there on Mars and didn't exercise vigorously to maintain his current muscle mass, similar to current astronauts spending time in the international space station.