  We provide a characterization of graded von Neumann regular rings involving
the recently introduced class of nearly epsilon-strongly graded rings. As our
main application, we generalize Hazrat's result that Leavitt path algebras over
fields are graded von Neumann regular. More precisely, we show that a Leavitt
path algebra $L_R(E)$ with coefficients in a unital ring $R$ is graded von
Neumann regular if and only if $R$ is von Neumann regular. We also prove that
both Leavitt path algebras and corner skew Laurent polynomial rings over von
Neumann regular rings are semiprimitive and semiprime. Thereby, we generalize a
result by Abrams and Aranda Pino on the semiprimitivity of Leavitt path
algebras over fields.
