This paper designs a cooperative activity detection framework for massive grant-free random access in the sixth-generation (6G) cell-free wireless networks based on the covariance of the received signals at the access points (APs). In particular, multiple APs cooperatively detect the device activity by only exchanging the low-dimensional intermediate local information with their neighbors. The cooperative activity detection problem is non-smooth and the unknown variables are coupled with each other for which conventional approaches are inapplicable. Therefore, this paper proposes a covariance-based algorithm by exploiting the sparsity-promoting and similarity-promoting terms of the device state vectors among neighboring APs. An approximate splitting approach is proposed based on the proximal gradient method for solving the formulated problem. Simulation results show that the proposed algorithm is efficient for large-scale activity detection problems while requires shorter pilot sequences compared with the state-of-art algorithms in achieving the same system performance.