To apply large scale geometry to physics requires the assumption of some small scale structure of vacuum space, maybe an average state of dynamic structures. GR has ignored this because in 1916 there was no way to postulate physical properties of space. Next progress occurred in 1930's but not until the 1960 was there enough experimental data to start guessing about the structure of vacuum.

Einstein allowed something called an ether to exist in vacuum as long as it had no preferred direction of movement relative to earth. Now the concept of ether or aether is not necessary or helpful in describing properties of space. Virtual particle pairs are used instead to describe dynamics of vacuum.

Space has properties of light speed, Planck's constant, and constants of electricity and magnetism, governed by how energy is partitioned and contained in space. Virtual particles contain energy of vibration, rotation, and translation, but usually for flat space equal in all directions. ZPE is the same concept of energy contained in oscillators made of virtual particle pairs that almost become real, but lack sufficient energy.

From the constants it is possible to derive the Planck units, also the frequency and energy of oscillators. The Casimir energy is demonstrated to exist, leaving one branch of science with too much energy in vacuum to have nearly flat space. Another branch of science allows the vacuum energy to be partitioned into four types related to the four forces, with a balance of positive and negative curvatures to give large energy density and nearly flat space at the same place. FLRW objects to this, but other metrics are supportive of it, leading in other threads to proposals for modifying FLRW.

Equal partition in flat space suggests unequal partition in curved space and during propagation of transient events. The geometry does not describe partition or the total energy of different types. So your answer is no. The geometry is a net result of competing energies and forces.  The geometry does not establish or control the constants.

If natural laws reside in vacuum space, the vacuum energy may provide the power to enforce the laws. Part of the energy might be borrowed according to Heisenberg uncertainty, but must be repaid when the allowed time has expired. The constants that moderate the natural laws are a consequence of energy content in space and how it is partitioned among the four forces.

When a physical object passes through space it alters the energy content and partition balance for a while in local space. The object has both potential energy and dynamic energy which interact with local space to alter gravity and inertia.

The universe has some geometry because of the energy content and types of energy. Geometry of space is just a description of how light moves over long distances though space in which the energy is partitioned, maybe a balance of energy types in flat space or an excess of one energy type over another in curved space.   

You have not exactly defined what you mean by fundamental constants. However, the standard model has a number of fundamental parameters. They are the 9 Yukawa couplings, 3 gauge couplings, Higgs VEV, quartic Yukawa coupling of Higgs, three quark mixing angles and a phase, and moreover, because neutrinos have tiny masses and because one does not know whether neutrino masses are of the Dirac or the Majorana type, we do not exactly know how many more parameters will come from the neutrino sector. Then there exists a tiny strong CP phase.

Then there are  dimension full fundamental constants ħ,c,G. All these constants are fixed by experiments. And also there is a cosmological constant term denoted by ∧ which has a numerical value of 2.035 x 10-35 sec-2.

Geometry, can however, dictate the form of certain fundamental equations of Physics. For example inverse square laws are dictated by a spherical geometry. This is because the same amount of electric flux originating from a point charge passes through concentric spheres of different radius. The same argument can be applied to obtain inverse square nature of Newtonian gravity.

Einstein's view of gravitation is also geometrical. Presence of masses deform space-time geometry near a massive object and the second massive body moves along a geodesic spending least possible amount of energy. But the point to note is that even in this case, geometry dictates the form of a dynamical equation and associated fundamental constants are determined by experiments.