Yes, it's much simpler in 5e - they just have one system for contested checks, which is used any time any two creatures are using ability checks against each other - this is everything from insight and deception in conversation, to holding a door closed against forcing it open, and includes attempting to shove a creature down or away.

So, in the case of shove, you use the attack action (so that martial characters with extra attack invest the same as anyone else - one attack - and can still use their other attacks before or afterwards if they wish), in order to make a contested ability check against your opponent: you use your athletics, and the enemy can choose whether to use their athletics or acrobatics to contest your effort. In the case of a tie, the status quo remains unchanged - so a door held closed remains close, and a creature remains un-shoved.