Would the wood of a rotting log be considered an ecosystem?
Almost anything can be considered an ecosystem
In the grand scheme of ecology a rotting log may not seem that important however almost anything in the natural world can be considered an ecosystem. First of all it depends on if you're talking about macro or micro ecosystems.
In this case, the rotting log would be a micro ecosystem. The log would provide food, shelter and interactions among species and the environment which would make it an ecosystem. These interactions between abiotic (nonliving) and biotic (living) factors are crucial to any ecosystem.
It's important to remember that an "ecosystem" can be almost anything whether you're referring to the entire biosphere or the tiniest tree stump in a forest. You just need to be able to identify the interactions within that system.