I use the stationary bike in my apartment's gym to train in winter. The bike does not give me power data, but it does give me "calories burned per hour." (as a rate, not a count) Is there any way to use this to calculate my power output? I assume the machine is measuring my power, because I don't see how it could calculate calories burned otherwise.
Calculating calorie burn from power output is pretty straightforward, but the reverse simply cannot be done. The same number of calories can be burned by doing an hour of 30 second efforts and 2 minute recoveries compared to doing a one hour steady endurance effort. Beyond that, you also have the inaccuracy of the gym bike measurement which will be the biggest factor to consider. Any bike which does not display power in watts to you also doesn't measure it in a way that is accurate enough to be useful in the slightest when comparing to data from any other unit.
In short, you can use the calorie number to compare between workouts on the same gym bike to know that you're improving if you burn more calories in the same time at the same level of exertion, but nothing more than that.
Calories/hour is actually a power measurement (power is the rate of transfer of energy). So naively one could just convert from calories/hour to joules/second (watts). 1 kilocalorie/hour = 1.162 watts. (Remember ‘calorie’ in a diet or exercise sense is actually a kilocalorie.)
However, I strongly suspect that exercise bikes calorie rate readings are quite inaccurate. No inexpensive exercise bike is going to have a proper accurate power meter in it. I suspect that calorie rate is estimated from speed and current resistance level.
[Update] As pointed out in comments, the calorie burn rate is an estimate of calories used by your body, not an estimate of power output to the cranks.
You also do not know what time interval the calorie rate is calculated over, making the readings not very useful.