Case Study: Did I need to declare an emergency?
Before we delve into this one.  Let's be clear that it is much easier for us to sit at a computer with books in hand and review decisions made under urgent circumstances.

Technically, the airport manager was correct here because per 14 CFR §91.126(b)(1):


  "Each pilot of an airplane must make all turns of that airplane to the
  left unless the airport displays approved light signals or visual
  markings indicating that turns should be made to the right..."


There do not appear to be any caveats to that rule.

To un-bind a flight from this rule, one would need to take advantage of §91.3(b)


  "In an in-flight emergency requiring immediate action, the pilot in
  command may deviate from any rule of this part to the extent required
  to meet that emergency."


In my personal experience, I have seen plenty of pilots fly around the pattern in whichever manner is convenient to them; entering on right or left bases for final, approaching for VFR straight-in landings from 15 miles out, etc.  So far, I haven't seen any repercussions for these actions, which is probably why your instructor did not agree with the airport manager.

Another thing that shouldn't go without being said is that this situation was none of the airport manager's darn business.

By verbally acknowledging that there was not an emergency, your instructor placed a liability on himself.  §91.3(b) allows the pilot to deviate from a part 91 rule, but to my knowledge, there is no statute that actually requires recognition of an emergency to be verbally declared or announced under non-ATC-controlled VFR flight.

By leaving the question, "is there an emergency?", unanswered, there is a chance to retrospectively say, "yes, we had an emergency and that's why we deviated from the rule."  However, since he told the airport manager that it was only an urgency he kind of shot himself in the foot if someone were to decide to raise the right turn as a legal issue, but I would be very surprised if anyone does.

To summarize and give you a more direct answer.  


You had a valid emergency
You had no responsibility to verbally declare it
You did deviate from a legal regulation to meet the needs of that emergency 


Sounds to me like you've got a very competent instructor.

Edit: I came across this Letter of Interpretation, which is sort of related to your situation.