Do non-resident aliens pay taxes on stock market gains?

A quick search brings me to the IRS site:

The Taxation of Capital Gains of Nonresident Alien Students, Scholars and Employees of Foreign Governments

Conclusion

Nonresident alien students and scholars and alien employees of foreign governments and international organizations who, at the time of their arrival in the United States, intend to reside in the United States for longer than 1 year are subject to the 30 percent taxation on their capital gains during any tax year (usually calendar year) in which they are present in the United States for 183 days or more, unless a tax treaty provides for a lesser rate of taxation.

This assumes that such capital gains are not effectively connected with the conduct of a United States trade or business.

These capital gains would be reported on page 4 (not page 1) of Form 1040NR and would not be reported on a Schedule D because they are being taxed at a flat rate of 30 percent or at a reduced flat rate under a tax treaty..

So a non-resident alien student, in the US for more than 183 days in a calendar year would need to pay 30 percent on capital gains absent any specific tax treaty.