The water will evaporate and fly out of the bucket; the process will not stop until there is enough water vapor in the atmosphere that the vapor pressure stops the water from boiling further. If the scenario happens in a confined space, this has a chance of happening while there is still liquid water in the bucket, but if you're expecting the water in the bucket to provide an Earth-sized atmosphere of water vapor then there will be no liquid water left.
(On the other hand, if you get things right, you might get some water ice at the bottom of the bucket, which will have been evaporatively cooled when the faster molecules leave the surface, taking energy with them and thereby lowering the temperature of the liquid. The remaining solid will then sublime slowly until nothing is left.)
You can see this cold-boiling effect in practice in the YouTube videos Boiling Water Until It Freezes by Cody's Lab and Boil Water at Room Temperature! by Practical Engineering.