I\'m trying to monitor network activity on my iPhone from within an app I\'m developing. Does iOS support a netstat-like command or something similar that can tell me what
You can use burp suite for this purpose , By this you can inspect all the http data going through your phone. You have to install this in your mac/ubuntu/windows machine, and then create a proxy server with help of this tool and then modify your wifi setting in iphone/android phone to use your pc as proxy , and then it will capture all the input/output traffic.
Read full instruction here - http://www.engadget.com/2011/02/21/how-to-inspect-ioss-http-traffic-without-spending-a-dime/