This question reminds me of, many moons ago in a galaxy far far away, the third week of every third month, when the pointyheads who think "vega" is a real word used to start talking about "pin risk" [or "the gamma hammer", if they wanted non-derivative grunts who don't think in Greeks to shut up and listen].
The punchline being that it matters what options the punters have traded vs the banks. Whether they are long or short options (of either type) close to the strike close to expiry.
If the punters are net long of calls and/or puts versus their banks, then the banks are thus on the short side of these trades. If the market rises, the banks have to buy to maintain their delta hedge. If the market falls, then the banks will sell. Adding to pre-existing price pressures thus will tend to pro-cyclically encourage market volatility.
But if the punters are net short of options at that strike, then the opposite applies. Maintaining the hedge will cause the banks to buy the dips, and sell the rallies. Which will tend to suppress market volatility.
All of these effects are (hopefully) obviously most extreme if/when the date is close to expiry, and spot is close to the strike on a glut of options. But the effect then is conditional on the mix of options that investors have traded, versus their banks have to hold. This can then have opposing effects, on the incentive for/against volatile momentum effects in the underlying.