It's too complicated to answer with a definite yes or no. First starters, most people's spiritual path is tied to some religious practice that they have learned through cultural or educational means (i.e., "I'm from UK, but I meditate"). Even if they do not adhere to the religion, the influence is undeniable. What does it even mean to be spiritually enlightened? That's a very diffuse term. As a Buddhist, I of course look to Buddha as an example. That is myopic though. Every religion has leaders they perceive as enlightened, each with different attributes defining their schema of the spiritually awakened.
Enlightenment from my perspective can be accomplished (theoretically) without practicing religion, but it is dependent on religion to a large degree. Similar to how many people say "I'm a moral person without religion," while most of their moral framework is derived from their culture's religious background. Religion is heavily integrated into human civilization. Even if you don't believe in something, that does not free you from its influence.
For non-Buddhist religious peoples they would probably argue "no," because spiritual enlightenment comes from directly their higher power. If there is a god then that probably is true.