Short answer...negligibly. There are about 64,000 Calls in open interest across all contract months and about 56,000 Puts. We don't know how many of those are long or short. Lets err on the side that most of the Calls are long, say 70% and that most of the Puts are Short(meaning Bullish), say 70%.
That means that 120,000 contracts are Bullish which equates to 12,000,000 shares. Now lets really err on the side of influencing the market and say that the MMs were on the other side of all of those, which is not the case, but illustrative to your question. That means that the MMs buy/offset shares to hedge Calls. More often they just match up a seller for a buyer. (Others...this is just a silly exaggerated example to illustrate my point)
That's less than two day's avg volume. So as I said, even erring on the extreme side of the equation....negligibly...;-)