Name of the software that can package websites to install on smartphones ?

Suppose I have a website written in asp.net and the database is SQL Server, after I "Publish the web site" and I want to encode this code for smartphones (both android and Android operating systems) iphone) what software do I use? Note that I only packaged the code in "Publish web site" and the SQL Server data I still kept on the server (not packing the SQL Server data)

You can't publish a website to work as an app on android or iPhone. Furthermore the legacy asp.net (using ascx files) is very old. Instead use asp.net with MVC or better yet create a website with .net core 3.0.

At any rate, that will get you a website that can be loaded in a browser on android and iPhone.

If you need an app that is installed to android or iPhone through the app or play store, you need to look at using something like Xamarin.