The gravitational field is not SU(4), it is analogous to an SO(3,1) gauge theory, meaning that the connection is a relativistic rotation on the tangent space, it tells you how to rotate vectors when you go to point to point, and the symmetry group is the Lorentz group. The symmetry the gives rise to gravity is not a unitary transformation which tells you how to rotate fields internally, like the other forces. But keep in mind gravity is not exactly a normal spin-1 gauge theory like the other forces, it is spin-2, and it is described by a different kind of Lagrangian, which is first order in the curvature, not quadratic. There is no immediate relation between them.
There is no numerological reason for the 1,2,3 pattern in the standard model, and it has nothing to do with the space-time dimension. It is expected that at high energy, the theory turns into SU(5) or SO(10), neither of which has any simple relation to space-time dimensions.
In classical gravity, there is some relation between GR and gauge theory--- GR on a symmetric compactification produces abelian or non-abelian gauge theories. The closest thing to a relation between space-time dimension and the dimension of the spacetime is Witten's 1980s argument: to make U(1), you take a circle, and the symmetry of the space is U(1). To make SU(2), you take an ordinary 2-d sphere, and to make SU(3), you take a 4-dimensional space CP_2.
So that the minimal internal space you need has dimension 4+2+1=7, and together with the 4 dimensions of space-time, you get 11. This was an argument that 11 dimensional supergravity can reproduce the standard model.
This argument is just numerology, it doesn't work. In string theory, which is the proper way to understand 11 dimensional supergravity, you get larger gauge groups from compactifications, and from defect-orbifold walls, or from branes in a compactification. The gauge groups are larger than the minimum required for the standard model, which is good, because the can accomodate SU(5) or SO(10), which don't fit in this way as 11 dimensional geometric symmetries.
So, in a nutshell, no, there is no relation, but people were doing something similar to this kind of numerology in the 1980s. This type of numerology was recently resurrected in the "exceptionally simple theory of everything", which tried to shoehorn spacetime and internal symmetries into a simple group, with even less success.