Sounds workable in theory, but...
As already stated in other answers, it would be a code violation because it would be unsafe; electrical codes exist for good reason.
Going beyond safety issues, there are all sorts of reasons why this just isn't a workable idea.
Most kitchen outlet boxes are wired with two "hot" lines so that top and bottom outlets can each serve up to a 15A load, for 30A total at 120V right? WRONG! The two hot lines are on opposite phases; between them you actually have 15A capacity at 240V. Yes you normally use them as if it was 30A worth of 120V, but that's not what is actually going on, so you can't simply join them in the outlet box. If you attempted to join them in the breaker panel (VERY BAD IDEA) you'd theoretically have 30A capacity on the hot lines, but only 15A capacity on the one neutral feeding the box (all sorts of unsafe/bad/fire hazard). If you tried to gang up power from other outlets, you might as well just run a new 20A circuit.
As already stated, your apenas two options are:
- Exchange the appliance for one which can operate on a 15A circuit, or
- Run a proper 20A circuit to feed your microwave.