Can I connect a car subwoofer directly to a home power amplifier without crossover?

A car subwoofer like this; yes.

https://www.focal.com/sites/www.foc...o-performance-nouveau-2019-focal-psb200_0.jpg

A car subwoofer like this; no.

https://www.focal.com/sites/www.foc...amplificateurs-caissons-de-basses-ibus-21.jpg

Most likely, your subwoofer won't have an internal crossover (although it might), so unless your home amplifier has a built in crossover, it will produce midrange sounds as well.

The difference is often a lower impedance, 2 or 4 ohms vs the more common 8 ohm impedance used in home stereos. This can have an effect on the amplifier used, where lower impedance means more strain on the amp. (Also, your amplifier has a impedance dependent watt rating; it produces 800W under a specific set of conditions, ie. 800W (maximum) if the connected load is 8 ohms (nominal). Lower impedance, more watts (and more strain). Higher impedance, less watts - and less strain.

Also, the car cabin itself boosts lower frequencies a lot more than a living room, so the box volume (and the resulting bass reproduction) will be less optimal for home use.