The underwater propagation environment for visible light signals is affected by complex factors such as absorption, shadowing, and reflection, making it very challengeable to achieve effective underwater visible light communication (UVLC) channel estimation. It is difficult for the UVLC channel to be sparse represented in the time and frequency domains, which limits the chance of using sparse signal processing techniques to achieve better performance of channel estimation. To this end, a compressed sensing (CS) based framework is established in this paper by fully exploiting the sparsity of the underwater visible light channel in the distance domain of the propagation links. In order to solve the sparse recovery problem and achieve more accurate UVLC channel estimation, a sparse learning based underwater visible light channel estimation (SL-UVCE) scheme is proposed. Specifically, a deep-unfolding neural network mimicking the classical iterative sparse recovery algorithm of approximate message passing (AMP) is employed, which decomposes the iterations of AMP into a series of layers with different learnable parameters. Compared with the existing non-CS-based and CS-based schemes, the proposed scheme shows better performance of accuracy in channel estimation, especially in severe conditions such as insufficient measurement pilots and large number of multipath components.