The obsidian bracelet is broken. Can I still wear it?

That's totally fine. You can wear obsidian on a new bracelet in a boutique jewelry store. The following is an extended introduction of Obsidian:

Obsidian is a common black low-grade gem, also known as natural glass, which is a naturally occurring glass. It is caused by the rapid cooling and condensation of volcanic lava and its crystal structure. Obsidian usually appears on the periphery of lava flows because it cools fastest. Obsidian is usually black, but brown, gray and a small amount of red, blue and even green substances can also be seen.