According the the FAA:
A near midair collision is defined as an incident associated with the operation of an aircraft in which a possibility of collision occurs as a result of proximity of less than 500 feet to another aircraft, or a report is received from a pilot or a flight crew member stating that a collision hazard existed between two or more aircraft.
http://aviation.stackexchange.com/questions/3680/how-is-a-near-miss-defined#3681
The question specifically asks about FAA regulations, so I will try to provide specific citations.
Summary
For a commercial airliner (as the question asked), separation will usually be at least 3 miles laterally, or 1,000 feet vertically. In the enroute environment -- at higher operating speeds above 10,000 feet and based on the type of Radar and distance from the antennae -- a 5 mile rule is applied laterally. This is true in most but not all situations. There are exceptions: see below.
Note also the "or": it is allowed (and in fact rather common) for two jets to cross paths at the same moment, with one 1,000 feet directly above the other. The 3 miles is only required if the jets do not have at least 1,000 feet of vertical separation.
http://aviation.stackexchange.com/questions/2806/how-much-is-the-minimum-safe-distance-between-two-planes-in-flight
NOTE: The ATC Controller CLEARLY violated the FAA regulations (both laterally and vertically) about minimum separation with the orders he gave to the two aircraft.
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