How do I install a .apk on my Nexus 7 using the browser?
The first thing worth noting that your original attempt at pointing your browser to the .apk directly should work (see the comments starting with Stephen Schrauger's). It's not really clear why this would not be the case for you, but you may want to double-check your Downloads app (in the app drawer) and your pathing. It could be that it's "downloading" in the background but it happens so quickly that you do not realize it.

If for some reason that really isn't working, then you can try a potential workaround. This is pretty ugly, but it does work. I've tested this on my Nexus 10, for reference.

Create a folder on your laptop that contains your apk files (we'll call it apkfolder for the purposes of this example), then within that folder create an HTML file (we'll call it apkinstall.html) using a text editor. Since you have an Astro apk, use this as the content of the HTML file:

&lt;a href="Astro.apk"&gt;Install Astro&lt;/a&gt;


Where Astro.apk should be the actual name of your Astro apk file.

Now copy apkfolder onto the root of your Nexus 7's storage using Android File Transfer. Make sure that all the apks copy over as well as the HTML file. 

If you are using TextEdit on a Mac: TextEdit will attempt to treat things as RTF files by default, so even if you save it with a .html extension it may not work. One workaround I've found is to go to Format &gt; Make Plain Text before you save the file. You'll know you're in the correct format when the rulers and such disappear from the TextEdit window. Then you can save the file and make sure you use change the .txt to .html.

Next launch your browser, and point it to file:///sdcard/apkfolder/apkinstall.html. You should get a mostly blank page, with a link that says "Install Astro". Click that link to install the Astro apk from your storage, then use Astro to install all your other apps. If you're using Chrome, then you'll probably need to pull down the notification bar and tap the "Download complete" notification to actually kick off the install.

In the more general case, if you didn't have a file manager apk, you could instead add a bunch of link entries to the HTML file - one for each app - and install all of them that way. In your case, this shouldn't be necessary.