In your scenario, the batter (who becomes a runner upon hitting the ground ball to first) is out and the runner originally on first is safe because the force play had been removed upon putting out the batter. This article asking about a similar circumstance and this question about a batter being put out while the runner stayed on first also alludes to such.
This is addressed under Rule 5.09(b)(6) in the 2016 MLB Rule Book.
Any runner is out when:
- He or the next base is tagged before he
touches the next base, after he has been forced to advance by reason
of the batter becoming a runner. However, if a following runner is put
out on a force play, the force is removed and the runner must be
tagged to be put out. The force is removed as soon as the runner
touches the base to which he is forced to advance, and if he
overslides or overruns the base, the runner must be tagged to be put
The definition of "force play" and the example provided illustrate when the force is removed.
A FORCE PLAY is a play in which a runner legally loses his right
to occupy a base by reason of the batter becoming a runner.
Example: Man on first, one out,
ball hit sharply to first baseman who touches the bag and batter-runner
is out. The force is removed at that moment and runner
advancing to second must be tagged.
Note that the runner on 1st is forced to vacate his base, as addressed under Rule 5.06(a)(1):
A runner acquires the right to an unoccupied base when he touches it
before he is out. He is then entitled to it until he is put out, or
forced to vacate it for another runner legally entitled to that base.
When is another runner legally entitled to first base? When he hits a fair ball. This is addressed under Rule 5.05(a)(1):
The batter becomes a runner when: