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Benefitting from the unequaled precision of the pulsar timing technique, binary pulsars are important testbeds of gravity theories, providing some of the tightest bounds on alternative theories of gravity. One class of well-motivated alternative gravity theories, the scalar-tensor gravity, predict large deviations from general relativity for neutron stars through a nonperturbative phenomenon known as spontaneous scalarization. This effect, which cannot be tested in the Solar System, can now be tightly constrained using the latest results from the timing of a set of 7 binary pulsars (PSRs J0348+0432, J0737$-$3039A, J1012+5307, J1738+0333, J1909$-$3744, J1913+1102, and J2222$-$0137), especially with the updated parameters of PSRs J0737$-$3039A, J1913+1102, and J2222$-$0137. Using new timing results, we constrain the neutron star's effective scalar coupling, which describes how strongly neutron stars couple to the scalar field, to a level of $\left|α_{\rm A}\right| \lesssim 6 \times 10^{-3}$ in a Bayesian analysis. Our analysis is thorough, in the sense that our results apply to all neutron star masses and all reasonable equations of state of dense matters, in the full relevant parameter space. It excludes the possibility of spontaneous scalarization of neutron stars, at least within a class of scalar-tensor gravity theories.