No it won't. Actually, the first time I answered this question I got it right by saying NO. But the second time I answered this question and said YES, and I had such conviction about it that I couldn't understand how I had reached the wrong answer the first time. I also checked my answer with three other planetary astronomers and they agreed with the 'earth stays put' answer.
Having said that, we have to be real careful with our language. If you mean 'does the Earth stay absolutely stationary' relative to the lunar horizon and sky location, the answer is NO.
The moon is locked so that it presents the same face to the earth all the time during the lunar month. If you carefully use two balls and watch what happens to the earth as seen from the moon, you will discover that the earth must always remain in ABOUT the same spot in the sky all the time. It will grow slightly larger and smaller as the moon moves from perigee to apogee. Also, the earth will go through a complete set of phases each lunar month, with a 'full earth' happening when it is new moon, and a 'new earth' happening during full moon. The problem is that the Moon undergoes a motion called Libation which causes it to rock slightly back and forth relative to a line connecting the center of the Earth and the Moon. This libation effect, as seen from the moon, will cause the Earth to move slightly back and forth in the sky relative to a fixed azimuth and elevation point above the lunar horizon.
Also, because the orbit of the moon is tilted about 5 degrees with respect to the Earth's equator, from the moon there will be locations where the earth will slowly rise and set during the lunar month as seen from the surface.
So the bottom line is that the Earth does not remain perfectly stationary in the lunar sky from every point on the Moon, but moves in a rather complicated way depending on your location.
I would like to thank Rick Johnson of the Hyde Memorial Observatory for encouraging me to get the answer more consistent with the real world.
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