AR 607-1, Section 1-10, Paragraph j., Sub-paragraph (2) is likely the most relevant reg covering this.

AR 670-1
Wear and Appearance of Army Uniforms and Insignia

1-10. When the wear of the Army uniform is required or prohibited
j. Wearing Army uniforms is prohibited in the following situations:
(2) When participating in public speeches, interviews, picket lines, marches, rallies, or public demonstration, except as authorized by competent authority.