Yes, you can use natural stone in an outdoor application. Depending on the climate you are in, the thickness of the stone can be a factor in preventing cracking from freeze/thaw conditions. A problem may occur in colder climates when water collects and freezes which could cause the stone to crack.
There are generally less issues with cracking in warmer climates. For example, the Charleston, SC area (where Melcer Tile is located) is a warm climate and slate and travertine are safe to use.
Another principle to follow is to pair the type of natural stone you with the outdoor activity. For example, it is best not to use slick surface stone around a pool.
Sealing is also affected by whether the stone is indoors or outdoors. When using natural stone in outdoor spaces, you will need to re-seal your stone more often as an outdoor application of sealer might last 1/3 as long as an indoor application.
Specific jobs may have other limitations. Please seek the advice of a tile professional to help in your specific application.