As mention by Bruno Pereira in a comment on your question you need to create an application launcher for SQlite Database Browser.

To do this open a text editor such as Gedit and create a text document with the following contents:

[Desktop Entry]

Encoding=UTF-8

Name=SQLite Database Browser

Comment=Create, Edit and Modify SQLite database files

Exec=sqlitebrowser %u

Terminal=false

Type=Application

StartupNotify=true

MimeType=text/plain;

NoDisplay=true

Once you have done this name the file sqlitebrowser.desktop and save the file to ~/.local/share/applications

Once you have done this navigate to a .sqlite3 and right click on it, select Open with Other Application

Then select Show other applications and scroll down to select SQlite database Browser from the list and it'll now be the default application to open .sqlite files.