Optimism nevertheless appeared intact among traders as bitcoin zoomed to a record high above $63,000 ahead of Wednesday's trading launch of the Coinbase cryptocurrency exchange.
Dow Jones Industrial Average rose 30.53 points, or 0.1%, to 30,990.53, the S&P 500 lost 1.52 points.
Europe's broad FTSEurofirst 300 index added 0.79%, at 1,575.88. The pan-European STOXX 600 was up 0.8% after Swiss wealth manager UBS posted a surge in quarterly net profit.
U.S. lawmakers continued to negotiate over additional stimulus to help offset the economic impact of the pandemic while pursuing a stopgap government-funding bill.
Trump issued the gloomy warning in his first formal White House virus briefing since the end of April -- and the news sent Asian and European stocks sliding on Wednesday.