Can I apply and play for the US Army sports team, even though I have no experience playing sports in varsity at high school or in college?

If you are talking about All-Army Sports, not really. That takes experience and typically recruited for a scholarship, or already played at a competitive level in college.

However, every post MWR has a pretty extensive intramural sports calendar that includes basketball, flag football, softball, volleyball, soccer, pickleball, and sometimes tennis, bowling, darts and even e-sports.
And there's typically a post rugby team that plays other posts and colleges.
Most players on a post rugby team got to the Army with no rugby experience and learn there.
If you have an interest in a few of them, you could be busy all year.

Technically, no experience is required .. but if you don't have any, you'll probably not make the screening. You can apply for any number of sports on the All-Army Sports Team*, but just like a job, if you don't have a resume that catches the screener's interest, it's not going to go anywhere.

There are likely many other options available through your post's MWR program though.
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* https://www.armymwr.com/programs-and-services/sports-fitness/all-army-sports/applications

If you happen to be very well co-ordinated and extremely athletically inclined, sure you can play Army sports.. All you have to do is to demonstrate overwhelming ability to the team’s manager/coach.