Williams said that when rates and inflation are low, policymakers cannot afford to keep their "powder dry" and wait for potential economic problems to materialize.
"He's toeing the party line at the Fed, basically implying that an insurance rate cut is the right thing to do for the economy at this point in time," said Chris Zaccarelli, chief investment officer at Independent Advisor Alliance in Charlotte, North Carolina, referring to Williams' comments.
Before Williams' comments, stocks had been lower as shares of Netflix Inc tumbled 10.3% after the company's quarterly results, in which it missed targets for new subscribers overseas.
Losses in Netflix triggered a 0.9% fall in the communication services sector, which has been one of the best-performing S&P sectors so far this year.
"Earnings have met expectations, but companies are being cautious about future quarters, which is something that's not able to keep the S&P 500 above the 3,000 level," said John Augustine, chief investment officer of Huntington Private Bank in Columbus, Ohio.
The Dow Jones Industrial Average rose 8.61 points, or 0.03%, to 27,228.46, the S&P 500 gained 9.48 points, or 0.32%, to 2,993.9 and the Nasdaq Composite added 17.75 points, or 0.22%, to 8,202.95.
Among positive earnings reports, shares of Philip Morris International Inc climbed 8.9% after the tobacco company raised its full-year profit outlook, while shares of railroad operator Union Pacific Corp jumped 5.4% after the company's profit came in ahead of expectations.
International Business Machines Corp shares rose 4.9% as the company's quarterly profit beat on strong growth in its high-margin cloud business.
Morgan Stanley shares rose 1.4% after the bank posted a better-than-expected quarterly profit. The S&P 500 banks index was up 0.8% after three days of losses.
UnitedHealth Group Inc shares slipped 2.2% as the insurer said on its conference call that 2019 revenue would not hit its original target.
Profits for S&P 500 companies are expected to rise 0.6% for the second quarter of 2019, according to Refinitiv IBES data. Until Wednesday, there were expectations of a dip in earnings.
Advancing issues outnumbered declining ones on the NYSE by a 1.20-to-1 ratio; on Nasdaq, a 1.22-to-1 ratio favored advancers.
The S&P 500 posted 32 new 52-week highs and four new lows; the Nasdaq Composite recorded 56 new highs and 95 new lows.