I love it when I answer my own questions, especially when I don't understand why it worked! I have (I thought) identical versions and settings of Apache, PHP, and MySQL, including the default httpd.config files. I confirmed the httpd.config files were identical. In particular, that index.php was not specifically included in the DirectoryIndex setting. On the machine without WordPress, I could definitely specify a directory without a specific file and it would find index.php if it was there.
It turns out that on my other machine, with WordPress, it wasn't finding index.php files.
So, I edited httpd.conf (in the Apache/Apache2/Conf/ folder) and added index.php to the DirectoryIndex line. Just added it after index.html and one other that was there by default.
You need to stop and restart Apache for the change to kick in. Now WordPress is running great.
I hope this works for you other folk. As I said, I have no idea why one machine without index.php included on that line found it, but this fix worked for me.
Bradh