We study the percolation of strongly connected clusters (SCCs), in which sites are mutually reachable through directed paths, in systems with randomly oriented bonds by extensive simulations on hypercubic lattices from dimensions $d=2$ to $7$ and on complete graphs. Below the upper critical dimension $d_u=6$, critical SCCs exhibit a nontrivial fractal dimension $d_{\rm SCC}$ with a nonmonotonic dependence on dimension: $d_{\rm SCC}$ reaches a maximum at $d=4$, in contrast to the fractal dimension of ordinary percolation clusters, which increases monotonically with $d$. The SCC size distribution follows $\sim s^{-\tau_{\rm SCC}}$, where the Fisher exponent is consistent with the hyperscaling relation $\tau_{\rm SCC}=1+d/d_{\rm SCC}$ and, unlike $d_{\rm SCC}$ itself, increases monotonically with $d$. For $d\ge d_u$, mean-field behavior is recovered with $d_{\rm SCC}/d=1/3$, consistent with complete-graph results. However, unlike finite-dimensional lattices, complete graphs exhibit a double-scaling structure in the SCC size distribution: large SCCs are governed by the mean-field exponent $\tau_{\rm SCC}=4$, while small SCCs follow a distinct power law with exponent $\tau \prime =1$. At $d_u=6$, logarithmic corrections are also observed. These results show that SCCs remain well-defined fractal objects across dimensions, exhibiting both a nontrivial dimensional dependence and a nontrivial crossover from finite-dimensional to mean-field behavior.