It's possible since Elasticsearch has JDBC-driver: https://www.elastic.co/downloads/jdbc-client
You should create a custom-driver based connection in DataGrip.
Then you create a new Driver on the 'Data Sources and Drivers' dialog. Click on the '+' button on the upper left corner. Choose Driver.

Complete the new driver's options. Use downloaded JDBC drivers there.
Now you can create a new Data Source based on this new driver. Click on '+' button on the upper left corner of the 'Data Sources and Driver' and choose your new driver.
Complete the Data Source options. And finally choose which schemas you want to access on the 'Schemas' tab.