My mail icon got deleted from my iPhone. How to I get it back. I did not backup my emails in iCloud.

See the following from Remove built-in Apple apps from the Home screen on your iOS device with iOS 10 - Apple Support

Restore a built-in app that you removed

1. Go to the App Store.
2. Search for the app. Make sure that you use the exact name of the app. Find the correct name of built-in apps.
3. Tap to restore the app.
4. Wait for the app to restore, then open it from your Home screen.

Backups has nothing to do with it. Here is a screenshot from my App Store showing the app. Do you have any Restrictions turned on to prevent installing apps? Did you do a Spotlight search to see if the app exists on the device at all?

Spotlight search is the search you do from the home screen on the iPhone. You would either swipe downwards from the center of the home screen to bring it up, or you would swipe to the right to bring up the today screen, which also contains the search ability at the top of the page. Type Mail there, and then see what results you get. If Mail is still on the phone, then you would see where it would be located, if in a folder. If not, then you should get a hit from the App Store. I have no idea why you are not seeing the Apple Mail app.

Ewe8199 wrote:

Well now when I tried to ope it, it makes me sign in and then it tells me that the email account is already set up on this phone and sends me back to the screen with all the different email account options.

When you try to open what "it"? Are you attempting to install the email account in the Mail app? You could go to Settings>Mail, and remove the account that is presenting problems and add it back again.

FoxFifth's post includes a link on how to find the correct name.

Use the App Store app on your iPhone (i.e., don't do it using iTunes) and just enter the word "mail" -- Mail from Apple was the second result in that search when I tested it a minute ago.

Much easier to do directly from the phone. I just did a search on "Mail". The second result, as FoxFifth indicated, is the Mail app from Apple. Make sure that you use the one from Apple.

It shouldn't matter, you can download multiple apps that perform the same function. I have the Mail app still on my iPhone that I did that search with, and it came up just fine.