Is there any legal way to file a labor lawsuit outside the jurisdiction of the employer: Saudi Arabia to another country like US

There is no straightforward quick answer without additional facts. The general rule is that in order to file a lawsuit against a person or corporation, the court has to have personal jurisdiction. Personal jurisdiction is easier when the parties are residents of the same state, however, when the parties are not residents of the same state, then other considerations come into play. There are rules for filing a lawsuit against a foreign subsidiary in the United States. Are you resident in the US? Which state are you employed? In order for the foreign subsidiary to be sued in a state in the US, you have to show that the subsidiary has minimum contacts in the state where you want to file the lawsuit. There are tests for establishing minimum contacts for foreign subsidiaries. In New York for e.g., the following issues have a bearing, bank account in New York, employees in New York, leases in New York, public relations or publicity work in New York, sales in New York. The bigger question in any case is, did the foreign subsidiary avail itself of the opportunity to do business in a state that it canot complain that it is unfair to file a lawsuit against it in a particular state? That is not an easy question to answer without additional facts from you. I am a California lawyer and only used the New York situation to illustrate issues related to jurisdiction.