Generalized propensity scores (GPS) are often used when estimating the causal effects of a continuous exposure in observational studies. Existing approaches include using the estimated GPS a) as a covariate in the outcome model, b) for inverse probability of treatment weighting (IPTW), or c) in doubly robust (DR) estimators. However, the following limitations exist: a) either the GPS or the outcome model, or both, need to be correctly specified; b) both IPTW and DR approaches relying on weighting are sensitive to extreme values of GPS; c) assessing covariate balance is not straightforward. We propose an innovative caliper matching approach that uses GPS with continuous exposures. Under the local weak unconfoundedness assumption, we provide theoretical results of our proposed matching estimators and introduce new covariate balance measures under the matching framework. In simulation studies, our proposed matching estimator outperforms existing methods under settings of model misspecification and/or in presence of the extreme estimated GPS values. We apply the proposed method to Medicare data in New England (2000-2012) to estimate the effect of long-term PM2.5 exposures on mortality.