At the current stage Nutch is only responsible for crawling the web, meaning visit a web page, extract the content, find more links and repeat the process (I'm skipping a lot of complicated stuff in between, but hopefuly you get the idea).
The last stage of the crawling process is to store the data in your backend (ES/Solr are the supported data storages on the 1.x branch). So in this step is where Solr comes to play, after Nutch have completed its work you need to store the data somewhere to be able to execute queries on top of it: This is Solr job.
Some time ago Nutch included the ability to write the inverted index (as explained in the question), but the decision (also some time ago) was to deprecate this in favor of using Solr/ES (or any other storage that you can write an indexer plugin for). Right now the indexing plugins are plugable and you can write a plugin for any data storage that you want.
Summary: Nutch is a crawler and Solr is the search engine where Nutch stores the data that is crawled.