The 553 error usually means that you do not have permission to write files to the specified server folder. I see that you are publishing your project to the site's /public_html folder. On some hosts, this folder cannot be directly written to. Instead you upload to "/" (no quotes) which can be written to and is an alias or link to the real folder.
In any case, your best option is to publish your project to a local disk directory (see the Pages tab in the Publishing window). Once you have done this, you have 2 options.
1. Log into your Hostgator account and use their File Manager to upload the files and folders inside your local disk directory to the /public_html folder. Be sure to upload what is inside the local publish directory and not the local directory itself.
2. Use a 3rd party FTP program (such as the free Filezilla) to upload the published files.
Option 1 is probably your best bet, but will be slower. While using an FTP program will be faster, if you have never used an FTP program before, then you will have a bit of a learning curve.