Q:
Are you trying to use Aware to "CREATE" web service or "CONSUME" web service?
ASP.Net WebAPI is a full fledged platform to create a REST service. So, I'm not sure why you brought up ASP and Aware in the same sentence.
Regardless what language or platform the service was written, for you to consume that service via REST service, you need to write a Java plugin, register it with Aware and then use the plugin to communicate with Web service.
So, you need to focus on the Java plugin and not worry what the REST service was written in.