Why didn't he field the ball first and then let the umpire decide if it's fair or foul?

"Davis was holding a runner at first and shifting to cover the hole, believing it was unlikely that the right-hander at the plate, Brandon Phillips, would hit the ball down the line. He knew that the runner at third, Shin-Soo Choo, would be fast enough to beat his throw home on a softly hit ball.

When a grounder came his way, Davis ruled out throwing home to get Choo. He also decided against an aggressive attempt at a double play. On a ball like that, Davis said, stepping on first base would remove the force at second, meaning Choo’s run would count. Davis did not want to concede the run, because he knew the overpowering Reds closer Aroldis Chapman would come in for the bottom of the ninth.

As the ball skittered down the line, Davis said, it seemed to be bouncing foul. If it stayed on that course, the count would be 0-2 on Phillips. So, in far less time than it took to read about those options, Davis made the call: do nothing. Naturally, the ball stayed fair, skipping over the bag for the go-ahead hit. The Reds won, 7-4."