It appears that the cop was wrong.  I sent this email to the Arlington Police Department:

Two nights ago I was stopped going into my apartment at 1410 N. Scott St. by an officer who said that I had violated the law on my bike.  He claimed that by my passing his car on the right, I had committed an infraction worthy of a ticket.  I told him I didn't believe that was true, but he said we could settle it in front of a judge.  Obviously I did not want that.  However, after checking the Virginia DOT website, it appears your officer was mistaken about the law.  The Virginia DOT page says,

Bicyclists may overtake and pass another vehicle only when safe to do so. Bicyclists may pass another vehicle on the right or left, and they may stay in the same lane, change lanes, or ride off the road if necessary for safe passing. Please note that passing motor vehicles on the right side may be extremely dangerous if the motorist does not see the bicyclist and attempts a right turn.

  Seeing as we were both moving forward on a straight section on 15th St. just before Jerry's Subs and Pizza, there was no danger and I had plenty of room to pass on the right.  I don't appreciate being harassed by an officer who doesn't know the law.  The law to reference is Code of Virginia 46.2-907.

It's possible I'm misreading something, but it looks like I did nothing wrong.  Either way, I think the cop should have more important things to do than harass cyclists that aren't running lights or cutting people off.  I only started reading the actual law because of a discussion going around on the D20 list about a guy who was ticketed after crossing through a red light when no traffic was present.  His case is a bit more complicated.