ArcGIS Pro can open .sqlite files directly in the Catalog Pane TM in a folder connection, which is great!
Sqlite databases come with different names, the most prominent being .sqlite, .sqlite3, .db and .db3.
https://en.wikipedia.org/wiki/SQLite
You can test if a e.g. .db file is a sqlite database by checking the header for the string "SQLite format 3" as described here https://www.sqlite.org/fileformat2.html or check for the magic number described in the wiki article.
It would be great if ArcGIS Pro would check the afore mentioned files if they are sqlite databases and also open them if this is the case.