Looker does not support plugging-in Restful APIs directly. Like @brecht said - you will need to ETL your data into one of the supported databases, like Amazon Redshift, for instance.
I work at Alooma, one of Looker's many partners. We offer a fully-managed data-pipeline as a service, allowing you to quickly connect any data source to your data warehouse for BI and visualizations via Looker.
One of our many native integrations is with external RESTful services. After a quick configuration page, your data will be pulled via REST, then be automatically mapped and loaded to your DW. From there using it in Looker is as easy as clicking on the table in the UI! If you're interested, we offer a free trial - you can use it to check our REST to Looker integration.