We have seen that the grand disparity that was believed to exist between the way Nature works here on earth and in the heavens is not valid. The question remains, however, can we learn everything we need to know by investigating phenomena here on earth and extending that result to the Universe at large?

The answer must be no for the following reasons:

1) Who would have thought to look for a law of Universal gravitation without the precise measurements and detailed analysis of Brahe and Kepler? Cavendish's laboratory measurement of G was done in response to interpret  results obtained for the solar system.

2) Even if someone would have used the Cavendish apparatus to map out the gravitational force between two bodies, independently of knowing Kepler's results, would we be able to infer a complete understanding of celestial motion?

No. We know Newton's Law of Universal Gravitation is an approximation.
For example, there are certain aspects of Mercury's motion that can not be explained using the Newtonian form. The current explanation of Mercury's orbital motion requires General Relativity. In fact, General Relativity predicts that the path of a beam of light will be bent in a gravitational field. This effect is too feeble to see in a lab on earth. It was first observed by starlight being bent near the disk of the sun in a solar eclipse.

3) If we consider then the solar system to be our laboratory, is that a big enough laboratory to establish all that could be known?

The answer to this must be no too. In the 20th century, since Zwicky in the 1930's,
it is known that either the gravitational force deviates from Newtonian gravity at large distances, or that there is substantial dark matter in and between galaxies. The density of dark matter is so low that it has an imperceptible effect on small scale motions, like that in the solar system. The data seem to favor the existence of some very large amount of unknown, maybe even exotic  ( Is this the new celestial matter ?) type of matter.

4) Is the galaxy large enough as a laboratory to pin down all the Laws of Nature?

This seems to require a negative answer as well. There are structures that encompass groups of galaxies, and the non isotropy of the cosmic background radiation is a pattern on an extremely large scale. We have also seen that the luminosity vs distance plot for supernovae (SNe1A) suggest that the universe is accelerating in its expansion. This was the discussion about "dark energy" or the cosmological constant. This effect is not seen until we look out to red shifts > 1, or about 6 billion light years.

Sometimes features of the world are not visible unless we look on the large scale.
In fact, the most recent analysis from WMAP, using the angular spot size of the CMBR temperature fluctuations, fits a flat space scenario. Hence, ignoring local gravitational distortions of space-time the sum of angles in a triangle that covers most of the universe is 180 degrees!

5) If we could include the entire universe in our laboratory, would we have enough data to explain it all?