GoldMine's API options do not currently provide a RESTful API. Although I've "heard rumors" that such may be forthcoming, I have no idea what reality is.
The GoldMine documentation does include a 430+/- page PDF named GMPE_APIGuide_2018-1.pdf (where 2018-1 may be whatever version you are running) that documents the current API options.
Hope this feedback helps.
I am currently putting the finishing touches on my forthcoming GOLDHOOK product that will allow you to host a listener on your own IIS server that will allow you to POST/PUT/GET data to/from your GoldMine in the Contact, Calendar, History, and automated processes areas. If you'd like to be included in any future product informational e-mails/announcements, please DM me your info!