We study composite likelihood estimation of the covariance parameters with data from a one-dimensional Gaussian process with exponential covariance function under fixed domain asymptotics. We show that the weighted pairwise maximum likelihood estimator of the microergodic parameter can be consistent or inconsistent , depending on the range of admissible parameter values in the likelihood optimization. On the contrary, the weighted pairwise conditional maximum likelihood estimator is always consistent. Both estimators are also asymptotically Gaussian when they are consistent, with asymptotic variance larger or strictly larger than that of the maximum likelihood estimator. A simulation study is presented in order to compare the finite sample behavior of the pairwise likelihood estimators with their asymptotic distributions.