The Halacha is that a goy who keeps shabbos is chayiv misa a goy who learns torah is also chayav misa. This halacha is disscussed in the gemara is sanhedrid.
The reoson a goy who learns Torah is chaya misa can be found in a drasha. The passuk calls the torah a “Morasha” or inheritence(for lack of better word) the gemara tells us – al tikri morasha ela meorasa. The torah is a meorasa a women who is engaged but not yet married. She is set aside for her husband and only him. So to the Torah is set aside for the Jewish people alone.
Similarly Shabbos is also just for the Jewish people and not for the non Jews. There is a machlokes however what the nature of “Shavas” is. It either means regular shabbos like we have which is just for us or it may also mean take a day off from work. The reoson according to the latter sevara is that a jew’s purpose is to serve hashem and learn… a non Jew is supposed to work, contribute to society not to take off random days and so nothing – like every sunday 🙂 However I believe the common understanding is that the prohibition stems from the former reosons.
Additionaly the umos haolam have 7 mitzvos if they violate any of them there is a din of misa – that is the standered punishment. This is if they do not set up there own set of Dinim if they do maybe not. ALso not clear if dinim is on the 7 mitzvos or rile of law in genaral.
Ayein the gemara sanhedrin and the RAMBAM