Can distilled water in gallon plastic bottles expire?

There is no reliable expiration date for commercially bottled water, apparently because relevant authorities (such as the US FDA) have declined to scientifically test for it. Per bottledwater.org:

The U.S. Food and Drug Administration (FDA), which regulates bottled water as a packaged food product [in the US], has determined that there is no limit to the shelf life of bottled water ... Because it is packaged under sanitary, good manufacturing practices; is in a sanitary sealed container; and does not contain substances (such as sugars and proteins) typically associated with food spoilage, bottled water can be stored for extended periods of time without concerns.

As such, any expiration dates you see on bottled water are the result of either the manufacturer's idea of when you should change bottles (which might be wanting to encourage stock rotation) or because you live in one of the places where your local authorities set an expiration date on a political, commercial, or consistency basis.

Theoretically, since consumer plastic packaging is not 100% impervious, any bottled water will get contaminated by outside organisms eventually. But no study I've been able to find has good data on what "eventually" looks like, and even baseless estimates range from 2 to 10 years.