I'd like to take a mobile website and add it to the home screen of iOS, but when I touch it, it would open in its own app, NOT in the Safari app. Is there a service that would let me do this or a way I could manually do it myself, sort of like Fluid App?
If you are looking to make your own app using web technologies, there are some options. Look into Cordova and PhoneGap. There are similar alternatives for making apps for desktop operating systems as well.
While they may be easier to develop depending on your existing skill set, they may not feel as fluid as native apps. While some developers do a great job making HTML5-based native apps or even hybrids, often times they feel non-native to the user.
If you have an existing mobile-friendly website that you want to package into a native app, you can probably make a basic app with a UIWebView that merely displays the website. This will almost definitely feel like a website to the end user.