Is the Tower (Local) Controller responsible for "separating" VFR traffic in the pattern in the U.S. in Class D airspace?
Separation requirements depend on the class of airspace, per ICAO Annex 11.

Class D: VFR aircraft are not separated from any other aircraft. ATC only provides traffic information about VFR aircraft.

Note the separation can be done visually. Considering the size of the typical class D airspace, this is the usual method for multiple VFR aircraft in the area.

Once the aircraft get to the runway, the ATC regulation JO 7110.65 3−10−3 states that ATC is responsible for making sure an arriving aircraft doesn't cross the threshold before an aircraft ahead of it on the same runway is either landed and clear of the runway (or a certain distance past the threshold), or has departed and crossed the opposite end of the runway.

Also per 7110.65 2-1-1:


  The primary purpose of the ATC system is to prevent a collision
   involving aircraft operating in the system.


So while there may be no hard requirements for ATC to separate VFR aircraft, they will do their best to make sure that the VFR aircraft have the best chance of seeing and avoiding each other.