There is nothing wrong with binge-watching your way through the summer. There is not a better time of the year to do this.
Do not feel bad about the endless hours you will be spending this summer staring at a screen. You deserve it! You worked hard this past school year. With classes, clubs, homework, jobs, and trying to have a decent social life, downtime got pushed aside. Now that you have fewer obligations, do this for yourself.
This is your time to relax. People may tell you that you should spend less time watching TV and more time being outside or that you should be doing something else. You do not have to listen to them. As long as you are still taking care of yourself, watch as much TV as you want. You can spend your summer doing whatever you want.
Catching up may also improve your social life. You will finally know what everybody is talking about. Those references will finally make sense. You won't have to avoid all the spoilers anymore.
If you want to rewatch all 10 seasons of Friends, there is nobody stopping you. Do not let catching up on shows be a homework assignment. Enjoy it. Pick whatever pace you want to watch the shows and however far you get is what you have accomplished. Summer is about having fun and doing the things you do not have time for during the school year, not trying to get as much done as possible.
You are not the only one catching up on shows. There are so many shows that it is nearly impossible to ever catch up on everything you want to watch. Take it an episode at a time and savor the plots and character development. I know I will.