I want to know if it's proper for a Jew to spit out bones or anything he doesn't want to swallow on top of the table he sits at.
Indeed there is a prohibition called Bal Teshaktzu, not to be disgusting.
The gemara in Chagiga 5a also refers to your case
Additionally, when Rabbi Yoḥanan reached this verse, he cried: “For God shall bring every work into the judgment concerning every hidden thing” (Ecclesiastes 12:14). [...] And similarly, Shmuel said: This is referring to one who spits in the presence of another and his friend is disgusted by his action.
More generally, one is forbidden to do things that others will find disgusting (see here for more details).