Success! For those who may run into this, here's the problem, and the solution:
Problem:
New Nexus 7 2013, running Garmin Pilot, cannot connect to Garmin GDL-39 ADS-B receiver.
Symptoms:
Nexus 7 appears to successfully pair with GDL-39. However, no data is streamed to the Garmin Pilot app. No GPS, or ADS-B data, is received.
In other words, it's paired but not really.
Solution:
1. Go to Google Play store, and download the free Garmin GDL-39 configuration app. IGNORE THE ADVICE THAT SAYS "YOU DON'T NEED THIS APP IF YOU ALREADY HAVE GARMIN PILOT". You need it.
2. Pair the Nexus 7 with the GDL-39. It really IS paired, it just can't communicate with the app, until you...
3. Using the configuration app (which inexplicably CAN communicate with the GDL-39 via Bluetooth, even though the app itself cannot) download and install the latest firmware into the GDL-39. They are at Ver 2.8. Mine was still on Ver 2.2. (Note that you must be connected to WiFi to download the latest firmware with this app.)
4. Once complete, force stop Garmin Pilot, and cycle the power on/off to the GDL-39.
5. Restart both. In a minute or two, it will automatically connect without touching anything at all.
I also had to go to my subscription page (in the Garmin Pilot app) and let it find my account before a connection could be fully established.
Note that you MUST have WiFi to receive/install the firmware update. Since my GDL-39 is semi-permanently installed in my plane, and it requires ship's power to work, that meant setting up a WiFi hotspot at my hangar.
Whew, what an ordeal. Moral of the story: Keep your firmware up to date! Just cuz it worked flawlessly with the old tablet doesn't mean it will work on the new tablet -- unless you've got the newest firmware version installed in the GDL-39.