Songs like "You Make me Feel Like a Natural Woman" and "Kisses Sweeter than Wine" use this phrase. "I begged her and I pleaded like a natural man..." in Kisses Sweeter than Wine.
In theology the concept of natural man seems to refer to a sinful nature not yet redeemed by God's grace. In the songs, the state of being natural seems to be a positive thing. The context suggests it means direct and free from affectation or pretention.