Not directly, no. However you can go to the Jobs page (Jobs link in the upper right), and then filter by saved search name, and then click the most recently run job. That will load the most recent results in the flashtimeline view.
Although it's not a single click maybe it's close enough for your purposes. I actually plan on building just such a view in the next few weeks.
Also if you're running 4.2 you might want to check out the new Alerts interface if you havent already.