Yes, you can connect the output signal to both tip and ring to put it on both stereo channels. Per the schematic here STUFF - LOOK MUM NO COMPUTER the output’s just direct from an op amp — not even a 1k current limiting resistor, surprisingly; I’d advocate adding that on both the tip and ring to make sure shorting one doesn’t short the other or the op amp, not that it’s likely to be a disaster if it does. Anyway, splitting an op amp output to two destinations is not a problem. Buffering the two separately would be nicer but not really required. People do this all the time, sending one signal to two destinations:
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