Which FAA regulations require compliance with TCAS II Resolution Advisories?
The requirement to respond to the TCAS II resolution advisory is somewhat indirect.  one key fact is that any TCAS installation has to be approved:  


  14 CFR §91.221   Traffic alert and collision avoidance system equipment and use. 
  
  (a) All airspace: U.S.-registered civil aircraft. Any traffic alert and collision avoidance system installed in a U.S.-registered civil aircraft must be approved by the Administrator.
  
  (b) Traffic alert and collision avoidance system, operation required. Each person operating an aircraft equipped with an operable traffic alert and collision avoidance system shall have that system on and operating.  


Key points here:  1. The installation must be approved.  2. If it's installed, it has to be used.  

From Part 121:


  14 CFR §121.356 Collision avoidance system.
  Effective January 1, 2005, any airplane you operate under this part must be equipped and operated according to the following table: {Detail omitted}  


This says anyone under Part 121 must have A TCAS installed and operating.
If you go back to Part 21, you can find that the installation approval requires a Type Certificate, Supplemental Type Certificate, or an Amended Type Certificate.

Part of getting that installation approval is a requirement for Airplane Flight Manual (AFM) or an AFM Supplement.  With regards to TCAS, the airworthiness approval is addressed by 


  AC 20-151B Airworthiness Approval of Traffic Alert and Collision Avoidance Systems (TCAS II), Versions 7.0 &amp; 7.1 and Associated Mode S Transponders  


Chapter 5 and Appendix A cover the requirements for the AFMS for TCAS.

While the AC is advisory, it is spelling out a means of complying with the rules, which in Part 23 (small A/C) and Part 25 (air transport) require a flight manual.

Elsewhere:


  14 CFR §91.9 Civil aircraft flight manual, marking, and placard requirements.  


requires operation in accordance with the limitations of the AFM.

In Part 121, airlines operate in accordance with their Op Spec, which will include similar rules.

And there is specific wording to address clearances and response to TCAS:


  14 CFR §91.123  Compliance with ATC clearances and instructions.
  (a) When an ATC clearance has been obtained, no pilot in command may deviate from that clearance unless an amended clearance is obtained, an emergency exists, or the deviation is in response to a traffic alert and collision avoidance system resolution advisory. However, except in Class A airspace, a pilot may cancel an IFR flight plan if the operation is being conducted in VFR weather conditions. When a pilot is uncertain of an ATC clearance, that pilot shall immediately request clarification from ATC.
  (b) Except in an emergency, no person may operate an aircraft contrary to an ATC instruction in an area in which air traffic control is exercised.
  (c) Each pilot in command who, in an emergency, or in response to a traffic alert and collision avoidance system resolution advisory, deviates from an ATC clearance or instruction shall notify ATC of that deviation as soon as possible.  


This basically acknowledges that pilots will respond to an RA and that they should report it as soon as they can after the fact.