Are you paying him to play on your team? Sounds more like the other captain doesn't want you to have a college player on your roster. There's no NCAA/ITA rule that would prohibit it but if the player is in-season he may not be allowed to play due to a team or department policy e.g. a lot of teams don't let players participate in intramurals because risk of injury.
Ah, but there is an NCAA rule against it - at least for D-1 and D-2. The D-3 manual isn't so specific in naming tennis.
From the D-1 manual:
17.22.8.1.1 During Academic Year. A student-athlete in tennis who participates during the academic year as a member of any outside tennis team in any noncollegiate, amateur competition (e.g., tournament play, exhibition games or other activity) except while representing the institution in intercollegiate tennis competition shall be ineligible for intercollegiate tennis competition unless eligibility is restored by the Committee on Student-Athlete Reinstatement (see Bylaw 14.6.3 for exceptions). (Revised: 1/10/91 effective 8/1/91, 1/16/93, 10/3/05)
17.22.8.1.1.1 Vacation-Period Exception. A student-athlete may compete outside the institution's declared playing and practice season as a member of an outside team in any noncollegiate, amateur competition during any official vacation period published in the institution's catalog.
From the D-3 manual:
14.7.1 Outside Competition. A student-athlete becomes ineligible for intercollegiate competition in that sport until eligibility is restored by the Committee on Student-Athlete Reinstatement if, after enrollment in college
and during any year in which the student-athlete is a member of an intercollegiate squad or team, he or she competes or has competed as a member of any outside team in any noncollegiate, amateur competition (e.g.,
tournament play, exhibition games or other activity) during the institution’s intercollegiate season in the sport (see Bylaw 14.7.3 for exceptions and waivers) unless restored to eligibility before that time by the Committee
on Student-Athlete Reinstatement. (See Bylaw 14.1.12, for student-athletes participating in a recognized foreign exchange/study abroad program). (Revised: 1/16/93, 1/11/94, 1/13/03 effective 8/1/03, 12/20/04, 1/10/05, 1/19/13