We use measurements of the galaxy-cluster angular size versus redshift to test and compare the standard model ($Λ$CDM) and the $R_{\rm h}=ct$ Universe. We show that the latter fits the data with a reduced $χ^2_{\rm dof}=0.786$ for a Hubble constant $H_{0}= 72.6_{-3.4}^{+3.8}$ km $\rm s^{-1}$ $\rm Mpc^{-1}$, and $H_{0}$ is the sole parameter in this model. By comparison, the optimal flat $Λ$CDM model, with two free parameters (including $Ω_{\rm m}=0.50$ and $H_{0}=73.9_{-9.5}^{+10.6}$ km $\rm s^{-1}$ $\rm Mpc^{-1}$), fits the angular-size data with a reduced $χ^2_{\rm dof}=0.806$. On the basis of their $χ^2_{\rm dof}$ values alone, both models appear to account for the data very well in spite of the fact that the $R_{\rm h}=ct$ Universe expands at a constant rate, while $Λ$CDM does not. However, because of the different number of free parameters in these models, selection tools, such as the Bayes Information Criterion, favour $R_{\rm h}=ct$ over $Λ$CDM with a likelihood of $∼ 86\%$ versus $∼ 14\%$. These results impact the question of galaxy growth at large redshifts. Previous work suggested an inconsistency with the underlying cosmological model unless elliptical and disk galaxies grew in size by a surprisingly large factor $∼ 6$ from $z∼ 3$ to $0$. The fact that both $Λ$CDM and $R_{\rm h}=ct$ fit the cluster-size measurements quite well casts some doubt on the suggestion that the unexpected result with individual galaxies may be due to the use of an incorrect expansion scenario, rather than astrophysical causes, such as mergers and/or selection effects.