The planes were no closer than 4.85 miles apart. They were on nearly parallel, but opposing tracks, almost 5 miles apart. They were not on anything remotely close to a collision course. The TCAS alert was because the two aircraft slightly encroached on separation requirements of 5 miles lateral, or 1000 feet vertical.
The pilot does need to react immediately to TCAS alerts, and it is certainly better to react excessively than insufficiently. But, he might have been a little... enthusiastic with this maneuver.