As you are likely to be aware, a Bounce is where someone visits a single page on a website and then leaves. Often, this is perfectly normal. For example:
Example One: A browser visits a One-Page website. By design, all of the information is on one page where the visitor does not have any options to click through to any other page;
Example Two: Let's say a browser performs a search, looking for a Restaurants' Contact Details. The Search Results bring up URL
www.example.com/contact-us. The visitor clicks through to the page and then fills in the forms and leaves. There is no need for the visitor to click through to other pages as they have found the page they are looking for. Some would argue that a low Bounce Rate, is this instance, would not be a great thing.
As such, it would not make any sense for search engines to include a website's Bounce Rate into their Ranking Algorithms.
On the other hand, there is something known as 'Pogo Sticking'. This is where a visitor performs a search query and then clicks through to a website. They then immediately click back to the Search Engine Results Page (SERPs) and start to click through to other website pages within the same SERPs.
Unlike Bounce Rates, 'Pogo Sticking' could be a good indicator to search engines that the existing entry into the SERPs does not adequately meet the needs of the searcher or their search query. As such, a search engine may allow 'Pogo Sticking' to affect how a web page appears within the SERPs.
Google does not rely on Google Analytics installed on websites. They have their own means of calculating the behaviors and activities of websites that appear in their search engine.
Helpful Resource: https://support.google.com/analytics/answer/1009409?hl=en