The media guide for the United States Women’s National Team only notes one doubleheader* with the Men’s National Team: a pair of matches held at RFK Stadium in Washington, D.C., on May 30, 1998, in which the women played New Zealand and the men played Scotland. Contemporary news reports in 1998 state that this doubleheader was the “first-ever” (SoccerAmerica) and the “[f]irst time both the U.S. men’s and women’s national teams have played a doubleheader (Baltimore Sun).**
But, that would not be the first time the men’s and women’s national teams had played at the same stadium on the same day.
Photo: The Yale Bowl, in 2005 (via ToddC4176/Wikipedia)
The first time was seven years earlier, on June 9, 1991, at the Yale Bowl in New Haven Connecticut. The USWNT played a “select amateur team” prior to a game between the USMNT and Italy’s Juventus. The New York Times covered the matches, both with a preview and a follow-up article:
NY Times – U.S. Women’s Team May Be World’s Best (Sunday, June 9, 1991):
The ball is round, and cannot be touched with the hands. Soccer has another rule, unwritten but tradition-bound: Teams from the United States make accommodating doormats for the world’s more talented feet.
Yet on Sunday at the Yale Bowl in New Haven, local fans will have the chance to see an American squad that is one of the top soccer teams in the world. Before the United States men’s national team faces Juventus of Italy, (minus World Cup players Salvatore Schillaci and Roberto Baggio), the United States women’s national team will play an exhibition against an amateur squad.
NY Times – Bane of the International Game: No Goals (Monday, June 10, 1991):
In an earlier match, the United States women’s national team dominated a select amateur team, 2-0. KRISTINE LILLY of Wilton, Conn., scored in the 27th minute. The second score was an own goal in the 40th minute. The United States already has earned a berth in the first women’s World Cup in China this November.
As for the USMNT, they were able to hold Juventus to a 0:0 draw, although a Juventus shot that may have crossed the line was not called. Official attendance, at least for the men’s match, was 33,547.
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NOTES
*Definition of “doubleheader:” In soccer, the term “doubleheader” has a slightly different understanding than in baseball, which is from where that word is borrowed. For soccer, a doubleheader can be defined as “two matches, each with different teams, played on the same day at the same venue, usually back-to-back with an intermission in between, and usually with admission covering both matches.”
In the USA, the term “doubleheader” has been applied to groups of soccer matches going back to at least 1921 and appears in dozens of newspaper articles prior to 1991. Here is a partial screenshot of a Google News (Archives) search for “soccer doubleheader:”
**The “first-ever”/”first time” descriptions for the 1998 doubleheader can probably be ultimately attributed to U.S. Soccer press release(s). Perhaps U.S. Soccer’s media department failed to do any research at that time. Or, if they did check for past doubleheaders, since the 1991 matches were not true international matches, the information about them is probably lacking, so overlooking those matches could have happened rather easily.
Also, it is likely that in 1991 U.S. Soccer did not consider the two matches a doubleheader as the women’s team was little known. So, instead of one half of a doubleheader, the women’s match may have been considered a sideshow to the main event: the U.S. Men versus Juventus.
Women's Soccer Express 
