Both Apache and PHP are 32 bit versions - in both cases they are the latest versions that are supported on my system.
The php.ini resource limits are exactly as David listed. Apache is exceeding those limits so that presumably eliminates PHP as the problem.
The browser isn't requesting the resources (at least not after the initial call - since I closed the browser and Apache continued to use more and more resources without any browser running at all.
All of the PHP scripts I wrote myself all run fine - it seems to be something specific to the Wordpress install process.
I managed to work around the problem by installing Wordpress on my web hosting and then FTPing the files to my computer. It now loads without Apache going into an infinite loop.
Thanks for all the suggestions David and Remon.