He will be able to compete but it's a matter of if he should or not. Concussion protocols in NASCAR aren't up to the level of other sports like the NFL or MLB, and there's an unfortunate chance they'll clear Earnhardt to draw viewers even though, health-wise, he shouldn't be anywhere close to the race track.
Although he has suffered from a serious concussion, he will recover. As long as he takes care of himself and does not experience any other setbacks, there is no reason why he should not be able to compete in Indianapolis. He will need to be careful and take precautions during his recovery, but he should be fine.
The Indianapolis 500 is one of the most important races of the year. As one of the country's top drivers, Dale Earnhardt Jr. will do everything in his power to ensure he is fit and fully recovered in time for race day. With access to some of the top doctors and specialists in the nation, the NASCAR driver will use his resources to make it to race day in a fully-recovered state.
Earnhardt Jr. might feel he is ready. His medical team might feel he is ready. The final call is up to NASCAR and the team's owner. They may receive the okay from his doctor but he was in a major accident and that sort of occurrence can really do a number on a driver's head. They may hold him out for the rest of the season if they feel it is necessary.