In my case, I was also not able to stop the Sitecore XConnect Search Indexer service of my Sitecore 9.0.2 instance. Remember, the Sitecore XConnect Search Indexer service is very closely related to the Solr cores. (If the Indexer service is not running, the effect will be that you will not find any contacts (even anonymous/known) getting listed in the Experience Profile when any visitor visits your website (Or) you cant delete any contact from any list in the List Manager)
When I checked the Solr service, I found that it was in the Paused state, I was not able to stop this Solr service. Then, I realized that the problem was with this Solr. This Solr service was running in the same port that the other Solr Service was running. So, I stopped the other service, then I resumed this Solr Service (that my Sitecore instance is now using). Then, I was able to stop/start the Sitecore XConnect Search Indexer service as usual.
Two things to remember always:
1. the Sitecore XConnect Search Indexer service is very closely related to the Solr cores.
2. Never create a Solr service with the same port number specified for another Solr service. Or the best recommended way is that you give a Service name along with the port number for avoiding confusions, for example, SOLR721HOST8985
Both services were of the same Solr version 6.6.2