The choice to apologize first is often the choice that saves the friendship.

Going first feels like conceding the argument, which is precisely why both people wait. What it actually does is end the standoff, and the standoff is usually causing more damage than whatever started it. Apologizing for your part does not require agreeing that all of it was your fault, and friendships are lost to the waiting far more often than to the original offence.

Send the short version to somebody you have been in silence with: what you did, that you are sorry, nothing else. Leave out the clause about what they did. The rest can be sorted once you are talking.