Web apps on OSX?

There's no built-in OS X behaviour that exactly matches that of iOS, but you can use what's often referred to as a site-specific browser to turn sites into OS X apps.

Fluid

Fluid lets you turn sites into single apps that appear like any other OS X app — in their own windows, with separate dock icons. Pretty useful for web apps like Gmail, but you can use for on pretty much any site. It's free, but for $4.99 you get some extra features.

Mailplane

Like Fluid, but specifically for Gmail. Since it's specific to Gmail, it's a bit more finely tuned, with toolbar buttons for the usual sorts of mail app actions. $24.95, with a 30 day free trial.

Automator

If you don't want a third party app, you can make a very basic site-specific browser using Automator. There's a good article from Andy Ihnatko on the subject, but the short version is:

In Automator, create a new Application.
Add a Get Specified URLs action, and enter the URL of the page you want.
Add a Website Popup action.
Save it.

That will give you a standalone app you can run to get a browser for a specific site. But it's quite limited compared to solutions like Fluid or Mailplane — as the name suggests, it's more of a popup than a standalone app.

Mozilla Labs Prism

This is my personal favorite and I've been using it for a while. Yes, it is an "inactive" project, but it still works well. It has a simple, easy to use interface:

I put up a blog post on how to use Mozilla Prism, in which I go into more detail about the various features.

Two successors to Prism

Mozilla Labs Chromeless

Mozilla Labs: Prism

This project is considered inactive.

Chromeless | Mozilla Labs comprises two 2011 blog posts: - Prism is now Chromeless - Webian Shell: A full screen web browser built on Chromeless

– and whilst Webian Shell may be of interest to readers, it's not an answer to the question.

To a 2012 question, Chromeless “inactive”?, Michael replied:

… The ideas and code for Chromeless have grown into the HTML5 Applications project. You can read all about it at https://wiki.mozilla.org/Apps …

WebRunner

Prism – MozillaWiki refers to Chromeless and WebRunner.

From Salsitistas - Salsita Software:

… Matthew Gertner CEO and Founder

… Before Salsita, Matthew led the development of a Mozilla Labs project called Prism (later WebRunner), a "single-site browser" that enabled users to turn web apps into desktop apps …

http://www.salsitasoft.com/webrunner/ no longer presents the WebRunner page. There's a February 2011 copy in the Internet Archive Wayback Machine.

Natifier does precisely this, for any app, for free, open source.

https://github.com/jiahaog/nativefier