Newtonian space Often, gravity in relativity is depicted as and the orbit of a celestial body around another body like a ball spinning in a deep bowl, which is said to be space influenced by the presence of a large mass.  But could not also Newtonian mechanics be used to make the same analogy (although neither Newton nor anybody else did so)?
Would Newton´s ball not spin fine in a curved surface?
 A: You could use the  same analogy of the Sun sinking into a grid lined floor, for both theories, but the Newtonian view of space and time was shown to be wrong in at least three different ways

*

*Newton's law of gravity implied the immediate effects of an action at a distance, as I'm sure you know, this means that if the Sun disappeared now, the Earth would immediately  fly off at a tangent, instead of the 8 minutes it really would take for us to detect anything. GR resolved this by combining space and time into spacetime


*The Sun suddenly vanishing is not a very likely situation, but what really stumped 19th century astronomers was that their predictions of Mercury's  orbit were not in agreement with experimental results. Newtonian mechanics could not account for this, but GR could, by taking into account the non-absolute nature of space and time.


The theory of relativity predicts that, as it orbits the Sun, Mercury does not exactly retrace the same path each time, but rather swings around over time. We say therefore that the perihelion -- the point on its orbit when Mercury is closest to the Sun -- advances.
In the diagram shown here, the amount of the advance is greatly exaggerated. The actual advance is only 43 seconds of arc per century.



*There are other effects, such as Deflection of Starlight, which was predicted by the earlier theory, but with an incorrect estimate of the deflection.


The first observation of light deflection was performed by noting the change in position of stars as they passed near the Sun on the celestial sphere. The observations were performed in May 1919 by Arthur Eddington, Frank Watson Dyson, and their collaborators during a total solar eclipse. The solar eclipse allowed the stars near the Sun to be observed. Observations were made simultaneously in the cities of Sobral, Ceará, Brazil and in São Tomé and Príncipe on the west coast of Africa. The observations demonstrated that the light from stars passing close to the Sun was slightly bent, so that stars appeared slightly out of position.

A: The short answer is yes, sort of, but the reason is probably more subtle than you're thinking (i.e. it doesn't just follow from some physically intuitive argument).
You see, there is a mathematical framework known as Newton-Cartan theory, developed by Cartan and Friedrichs in the 1920s, in which Newton's theory of gravitation is re-expressed in the language of differential geometry in a manner very similar to that of General Relativity. 
In this representation of Newton's theory of gravitation (which gives precisely the same empirical predictions as Newton's theory) we have the Equivalence Principle implemented through geodesic trajectories defined in a curved spacetime, which itself has a structure influenced by the matter in it, just like Einstein's theory. 
In fact, this reformulation of Newton's theory provides, in my opinion, the most compelling means of deriving Newton's theory of gravitation as a limiting case of the general relativistic theory.
It is in this somewhat sophisticated sense that what you say is correct: both theories (Newton's and Einstein's) have equal claim to the ball-and-bowl or ball-and-rubber-sheet analogies, but I agree with another commenter that these physical models are very poor analogues of curved spacetime.
A: The idea of a planet running around the surface of a bowl is often used to discuss planetary orbits in GR.  And the same analogy can be used to discuss orbits using Newtonian mechanics.  The real question here is what is the difference between GR and Newtonian mechanics.
When Newton wrote Principia he could not explain the action of the gravitational force.  In essence how does the Earth "know" that the sun is there to orbit around it.  He called it a spooky action at a distance and left it as an open question.  In the derivation of GR Einstein matched the stress energy tensor which models the matter and energy in space to the curvature of space-time.  It is difficult to visualize that curvature of space-time so we try to to give an example.  Because GR is a theory of potentials it is natural to describe this curvature in two dimensions in this way using the stretched sheet with a mass in the middle and the planet riding around the sheet.  The same example can be used to discuss Newtonian orbits, but there is no explanation as to why the surface is curved without GR.
