I don't know how to answer that question, but suppose we strengthen the hypothesis by making it apply also to all polynomial rings $R[t_1,\dots,t_n]$ over $R$, for $n \ge 0$. Then because $K_i(R-Mod) \to K_i(R[t]-mod)$ is an isomorphism (Quillen, Theorem 8, Higher Algebraic K-theory:I) for all noetherian rings $R$, it follows that $K_i(R) \to K_i(R[t_1,\dots,t_n])$ is an isomorphism for all $n$ --- that is called $K_i$-regularity of $R$. The paper "$K$-regularity, cdh-fibrant Hochschild homology, and a conjecture of Vorst", J. Amer. Math. Soc. 21 (2008), no. 2, 547–561, by Cortiñas, Haesemeyer, and Weibel available at http://www.math.uiuc.edu/K-theory/0783/chwpost.pdf shows that $K_i$-regularity of $R$ for all $i$ implies regularity of $R$ if $R$ is an algebra essentially of finite type over a field of characteristic 0.