If there is a need for the zone type, yes, abandoned buildings will eventually sort themselves out. If, for example, a high-density residential building becomes abandoned, the lot is free for new high-density residential development. However buildings that replace an abandoned one will start fresh at base level and need to work its way back up over time. Effectively no different than had you bulldozed it.
There is a downside to leaving them to sort themselves out - an abandoned building reduces the happiness values around it, has increased crime rate, and is a higher fire hazard. All of which can conspire to reduce the property values of the land they occupy and the surrounding area. This makes the land less desirable to develop, and can potentially cause surrounding areas to themselves become abandoned.
So it is typically the superior choice to manually bulldoze abandoned buildings than letting them sort out their own problems. If you are not terribly concerned about achievements, there are mods available on the Steam Workshop that will automatically bulldoze buildings that become abandoned or burn down. This can be a significant time saver when your city gets sufficiently large.