Is there an Android Bitcoin app that can interact with other Android Apps on the same device?
The answer is the integration-android subproject of bitcoin-wallet.

Start with the source repository for
the Bitcoin Wallet.

The source code for in-app payments is in the BitcoinIntegration.java file, which is under the integration-android directory.

Instructions for compiling this subproject and example source code for using it are given in this slide presentation by Mr. Schildbach

There is also a sample android app that demonstrates this feature.

My thanks to Andreas Schildbach for personally sending me the answer to this question!