R1 on first, batter hits grounder to 1B who tags R1 who is standing on first, and then 1B touches first base. Is this a double play
Until the out at first is completed the runner standing on first has no rights to the base.
MLB rule 7.01:
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.
If there is still a force in effect (the batter is running towards first and has not been put out), then the runner is not entitled to first base and can be tagged out. This is more specifically laid out in 7.08e
[The runner is out if 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 out. However, if the forced runner, after touching the next base, retreats for any reason towards the base he had last occupied, the force play is reinstated, and he can again be put out if the defense tags the base to which he is forced
1.03b is also relevant here:
If a runner is forced to advance by reason of the batter becoming a runner and two runners are touching a base to which the following runner is forced, the following runner is entitled to the base and the preceding runner shall be out when tagged or when a fielder possesses the ball and touches the base to which such preceding runner is forced
However, this is exceptionally rare in high level baseball as I cannot conceive of a situation where a runner would move backwards on a ground ball. An MLB runner will be 10-30ft from the bag when contact with the ball is made. He will then determined the kind of contact, if it's a ground ball he's going to run as fast as he can to the next base, if it's a liner he will freeze and if it's a popup or flyball then he will use his judgement to determine if a fielder has a play (go back to the bag and tag up) or does not have a play (advance quickly to the next base and beyond).
The key wording is 7.01 is "forced to vacate it for another runner legally entitled to the base." Look further into the rules and you will find that the batter is not legally entitled to first base until he reaches that base safely. Therefore, the runner that was already on 1st base is legally entitled to 1st base until the batter reaches the base safely. When this happens, the runner is now not legally entitled to 1st base and can then be tagged out if still standing on the base. Until that happens, the defense has to throw to 2nd base to create a force out and cannot tag the runner standing on 1st base.