Did he score or not?

I would think run counts.  This is a timing play and since the run crossed before the third out it counts.   Now... if the ball was a grounder to third and there was a triple play where the runner was out at first, then the run would not count.

Run should count because the third out was not a force play - it was a timing play. Once a batted ball is caught in the air, there is no "force play" possible.  

See 4.09 -->HOW A TEAM SCORES. 
(a) One run shall be scored each time a runner legally advances to and touches first, second, third and home base before three men are put out to end the inning. EXCEPTION: A run is not scored if the runner advances to home base during a play in which the third out is made (1) by the batter-runner before he touches first base; (2) by any runner being forced out; or (3) by a preceding runner who is declared out because he failed to touch one of the bases.

Approved Ruling: One out, Jones on third, Smith on first, and Brown flies out to right field. Two outs. Jones tags up and scores after the catch. Smith attempted to return to first but the right fielder’s throw beat him to the base. three outs. But Jones scored before the throw to catch Smith reached first base, hence Jones’ run counts. It was not a force play.

In your case, the batter was the first out, there were no force outs (runners not forced to advance due to the fly ball being caught), and the out was on a trailing runner - not a preceding runner.

Thank you gentlemen!  I don't like the result, but I accept the ruling and appreciate your opinions.

Always stinks when the answer isn't what you wanted. Just remember if the last out is a force or the BR doesn't make first, no run. Anything else is a time play, the other coach used the wrong logic too. He said because he tagged and scored, that run should count. The other half of that is he has to do so before the third out is made. Say he had gone partway home, returned and tagged, then scored. All that had to happen before the third out.