Is there a way to view running apps on the iPhone?

There is no such thing. Part of the development model is that there are no background processes allowed. Applications are allowed to be on or off. All activity indicators are to be pushed to the phone (new IM, search results done, etc.).

Going to the home screen quits an app (providing the app time to save its state). If there were an activity monitor, it would be horribly simple to look at: core OS functionality and the one app running (which would be the activity monitor; meaning – you guessed it! – no other apps running).

Sorry to bust your bubble, but you are looking for something that does not exist on (non-jailbroken?) iPhones/iPod touches.