To answer this question concerning the most popular sports in the US (Baseball, Basketball, Football, Hockey, and Soccer) it can largely be view by size and athletic ability.
This answer assumes the average woman to be smaller than the average man and the athletic abilities to correspond. It also applies to only these 5 sports.
Of the sports listed, Basketball and Football are most reliant on size. This chart shows the average size of players from various sports.
In those two sports, women would have basically no chance because of their limited size combined with the superior athletic ability of men. Except for a possible long shot of kicker/punter as @Joe explains.
This leaves us with the other three sports:
- In the NHL the only position that a woman would stand a chance physically is at goaltender. @jerepierre has already touched on this.
- In soccer the women could match up with the average size according to the business insider article, but the men would be superior in athletic abilities such as jumping and sprinting.
- Women would have the best chance in the MLB as a knuckle ball or junk ball pitcher. To do this they wouldn't necessarily have to be athletic or big. They would need to perfect their craft of being a junk baller. This would also obviously lead to other issues besides talent, but as John Kruk once said, Lady, I'm not an athlete. I'm a professional baseball player.
Obviously sports that combine size and athletic ability would be an advantage to men. This encapsulates most sports. The possible niche being a sport where technique and practice can over take. A good example is pitching in baseball.
I stated in a comment on @Coach-D's post that I played NCAA baseball and we used to scrimmage the women basketball team at our school. They had some players that were taller than our players, and they were more fundamentally sound than we were. We would regularly beat them, and beat them easily.
The thought that Mo'ne Davis could compete in the MLB with a normal skill set (by which I mean a fastball and two off-speed pitches) is absurd. Unless she developed a devastating knuckle ball she wouldn't stand a chance. This is purely physiological, females mature earlier than males and most likely we saw the best pure athlete that she will ever be. Don't get me wrong she could refine mechanics and control, but the likelihood she throws over 85 MPH ever is slim. If she were able to do that, then she would join the club of thousands of males in the minor leagues and independent leagues that can do so also.
I don't not consider baseball a sport where you play the ball and not the man, as hitting versus a pitcher that throws in the 90's vs the 70's is a significant difference. A sport that could fit into that is golf. The ball lies the same no matter whom is playing it. This leads to Annika Sorenstam who played with the men on the PGA tour and missed the cut in the Bank of America Colonial tournament. During the same year she was able to win on the LPGA tour, leading me to believe that even for a woman to make the cut on the PGA tour would be a significant accomplishment.