A player was involved in a rundown play between 2B and 3B. The fielder throws the ball and strikes the runner in the nose. The runner immediately falls to the ground. I , as the umpire immediately called time out and went to the rescue of the injured player. A defensive player picked up the loose ball and tagged the runner on the ground. I called the runner out as in my opinion, he was not going to get up and continue in the rundown. Did I do this properly? Thank you

IMO Randall, the second you called "time out" the play is now over and I would consider everything a "dead ball" ruling. I am sure there are way smarter men and women with rule specific data to back up what I am saying but while you are correct that the person more than likely wasn't going to continue the run down, because you called time you should have put the runner back to 2B. I mean what you are saying is that if I don't feel like running the guy down, I can simply peg him in the face and then tag him while he is lying on the ground and save myself all that running back and forth? :-)

(and yeah, just trying to be funny BUT you see what I am saying)

I've always interpreted this play as live and if the player cannot get himself to the base safely, he would be ruled out. I do agree, once the ump calls dead ball, the play stops and the ump has to determine what happens next.

Along this line... last year I was in a run-down formally known as a pickle or hot box; the fielder threw the ball relatively hard right at my crotch, I instinctively swung my hand down to save myself if you know what I mean. My hand hit the ball just in time and it went into right field; I walked safely to second base. The umpire called me out because he thought I swatted the ball away on purpose, which I did because the alternative seemed worse. I didn't argue, just laughed it off. I'm always grateful that we have people who come out to umpire which allows us to play, and we should not complain and just thank them for participating.

Randall, that’s a tough situation because most anyone with an ounce of sense would stop play and run over to the injured player. You calling time out would be the instinctive thing to do and I don’t believe you’re necessarily wrong but if I was umpiring I would not have called time out.
At the risk of sounding cold-hearted the reality is there wouldn’t be an immediate reason for an umpire to call time out on that play. Since players were likely surrounding him (action had stopped) there wouldn’t be a ball hit at him, thrown at him or the risk of him colliding with a defensive player. Even though he’s hurting he’s not in any danger. If the defensive team gathers the ball and tags him I have an out, irregardless if time was called or not.