How can I export a single google calendar event to an ics file?

1.
In the calendar list on the left side of the page, move your mouse over the calendar you'd like to export from.

 2.
You'll see a drop-down arrow appear next to the calendar name. Click the drop-down arrow and select Calendar settings. (Alternatively, click the drop-down arrow next to My calendars and select Settings, then click the appropriate calendar from the list.)

 3.
Click the ICAL button in the 'Private Address' section at the bottom of the page, and click the displayed URL.

 4.
Save the exported file to the desired folder on your computer. , possibly by pasting that URL into another browser window, copying the window text,and pasting it into an editor.

This exports an entire calendar, not a calendar event.

I think you're thinking about this the wrong way.

Sounds like you just want to share certain events with certain people.  Does adding their email to the guest list not send them a request that they can add to their own calendar?  I'm not 100% sure, but I think that has worked for me before.

If you just absolutely need an ical file with one event in it, then ProfessorJimJam's answer above is probably the easiest/quickest way to do this.

What I'm really trying to do is make certain events on a Google calendar show up on our website in a custom calendar app (really a calendar plugin on a Joomla-based site). The calendar plugin can import ical files. We have a Google calendar maintained by some volunteers that contains a mix of events, some private (like meetings) and some general interest.  It's cumbersome to select an event, export it, then import it - even more so to move the event from the calendar where it is normally maintained to another one (the single event one), and then export it and import it. If there's nothing available to do it, then we will go with the approach described by Professorjimjam.

I guess it's unreasonable to expect full functionality from a free sample app like Google Calendar if we paid for it, I would want a lot more connectivity features.

When editing an event, check out the 'Publish event' button at the very bottom of the page.

If you invite yourself via an external email address, and click to send an invite, you will receive an email with an ics attachment.

Add your google account to a program like Thunderbird, mac mail or outlook.  Then drag the event to a folder on your computer.  it will make an ics for for the single event.  There is no better way to do this from purely the web browser. Personally I would say the best thing to do for you is make a param a part of the event name or comment.  something like

Community Meeting #publish

then you can have a script that reads the private ics file for the calendar, parses events and only keeps ones with "#publish" in the event name or some such nonsense.  I have done this and It works flawlessly.  the ICS file format is super basic to manipulate.