This paper addresses the problem of detecting distributed targets in heterogeneous Gaussian clutter without assuming the presence of secondary data. Specifically, the clutter is modeled as a spherically invariant random process with unknown texture components and covariance matrix structure (CMS). In contrast to existing approaches that are based on the estimate-and-plug techniques, we introduce an approximation of the generalized likelihood ratio test that leverages an alternating estimation procedure to obtain at least a local likelihood maximum. We also prove that the proposed method achieves the constant false alarm rate with respect to clutter parameters when appropriately initialized. Finally, a comprehensive performance analysis is carried out by Monte Carlo simulation and in comparison with the non-scatterer density dependent generalized likelihood ratio test (NSDD-GLRT) in the cases of known and unknown CMS. The results show that the proposed solution is more robust and effective than NSDD-GLRT when the CMS is unknown while exhibiting only a modest performance degradation with respect to the benchmark when the CMS is known.