We consider logistic regression with covariate measurement error. Most existing approaches require certain replicates of the error-contaminated covariates, which may not be available in the data. We propose generalized methods of moments (GMM) non-parametric correction approaches that use instrumental variables observed in a calibration subsample. The instrumental variable is related to the underlying true covariates through a general nonparametric model, and the probability of being in the calibration subsample may depend on the observed variables. We first take a simple approach adopting the inverse selection probability weighting technique using the calibration subsample. We then improve the approach based on the GMM using the whole sample. The asymptotic properties are derived and the finite sample performance is evaluated through simulation studies and an application to a real data set.