Not exactly. A boat in the Kuwaiti desert means slightly higher sea levels, enough for the salt marshes seen up towards Iraq to be somewhat bigger in area, large enough to cover this now dry location. The area is hardly mountainous and sea levels fluctuate throughout history. The middle east was once a far wetter place than now, just as the American midwest was once tropical and had oceanfront property.
As for fossils on mountains, they are there because of geologic uplift, as can be seen by the sharp angle of the bands of fossilized sea bed. Clearly, when the bands were made, the area was flat seafloor, not mountainous, and the layers were made over a very long time period before the uplift began. The fossil sea critters now seen in mountain rock are of course far older than the boat in any case. Wrong time period to be concurrent with the boat, unless you're thinking multiple floods.
Seriously, most of Kuwait is the result of millenia of silt building a delta at the base of the river.