I am using ArcGIS Pro 1.0.0. Yes, it works, I tested it before posting the other day. I just tested it again, and it worked without any issues:
arcpy.MakeFeatureLayer_management(r"D:\tmp\test.sqlite\webMerc1","webMerc")
<Result 'webMerc'>
It is helpful if you can post specific code and error messages, especially since it keeps not working for you.
I am not sure what is different between our machines. I recently re-imaged my machine using Windows 7 64-bit Enterprise Edition. I installed ArcGIS 10.3 before I got around to putting ArcGIS Pro on the machine, but the installations should be compartmentalized from each other. Anyhow, I was loading SQLite databases into Pro with the beta versions and pre-releases before I re-imaged my machine.
Have you tried creating a layer file in ArcGIS Desktop that points to a SQLite data source and then load that layer file into ArcGIS Pro?
As has been pointed out, SQLite isn't officially supported yet, so the best approach in the interim might be to find an alternate data store.