Is it possible for our external Java application to make a call to Ignition to clear/ack an alarm?
If so, what is the proper way to connect to the gateway for this sort of action? I can only seem to find documentation available from the perspective of a module loaded into Ignition.
No, your external application cannot call any part of Ignition's API.
Only a module loaded into Ignition can do this.
However, Ignition does offer a module that permits you to develop your own web hooks, called the WebDev module. With it, you can create web API endpoints that call SDK APIs on your behalf.