I think that button is somewhat difficult in gmail/android, since the gmail implementation "caches" the emails locally, but really most of it is pulled dynamically from the web.
NOT what you are looking for, but a possible alternative for you, which revolves around labels in gmail.
Create a new label (call it anything you want), then create a filter that adds ALL incoming mail to that label (and leaves it in the inbox).
Add a google labs feature called "multiple inboxes" and make a second box below your inbox that searches for that label.
Now when mail arrives on your desktop and you have read it, simply hit archive. This will remove that message from the inbox, but it will still be sent to that label. When you are TRULY done with it, you simply hit remove label (one of the multiple inbox buttons available)
When mail arrives on the Droid, you can read it and hit archive. It will dissapear from the droid, BUT still be accessible from the droid in a pinch simply by opening that label.