A few problems with this...
1) Polls carry no legal weight. They are usually done by many different groups, sometimes with different agendas. You would have to make an official, government run (or at least one with mandated standards) poll in the situation you present.Then make it a legal vote. Or make it a licensed poll that carries the same weight. Doesn't seem like much of a challenge.
2) How the questions in the poll are worded counts, and this would be an incredibly contentious issue. Also, people's opinions are more nuanced than a simple approve/disapprove. If all we're asking is "should he go or should he stay?" I can't really see it being misinterpreted. Ask the question directly and clearly. Let people give their own reasons in a blank space.
Take health care reform. The number seen bandied about by most conservatives is 70% disapprove of the healthcare bill (I count myself as one of them). However, when you parse the numbers, you find that the majority of people who disapprove of the health care bill do so because it doesn't do enough to change the system, not that it does too much. In this instance, it is incredibly disingenuous for the Repubs to use that figure as if it plays to their advantage. I disapprove of Obama now, but my disapproval and the disapproval of a teabagger are much different things.So then isn't it better just to scrap the whole thing and start over at this point?
3) When is the last time you took a poll? I see poll numbers bandied about all the time, but I can count on one hand how many polls I've participated in in the past five years, and I suspect at least two of them were push polls. There would have to be much wider participation in polls for them to truly represent the country, and that opens them up to much more fraud.Do we have clean elections? Can we afford to have them?
4) Even if you resolve all those issues, you have the problem of stability. It seems quite easy for a politician to come under 50% approval rating depending on the time. Would a recall be triggered the moment they go under that level, or if they are under it consistently for a period of time? Regardless, it would add much more instability to the system if the possibility of recall elections was ever present.It might be prudent to require that a few levers be pulled before a recall goes into effect. We can make the system as slow or as fast as it needs to be. I would not mind a little instability if it meant an emerging police state could be easily toppled.
5) This would also dumb down politics. Since poll numbers are heavily influenced by public perception, it would make presidents favor flashy "bread and circuses" type stunts over sound legislation.Politics is already dumbed down. It's not going to go much further. Besides, I'd rather have bread and circuses than war.