You cansue the Marine Corps for a decision to discharge you, but you will not be successful.
Contact your local Veteran's Affairs Office. A dishonorable discharge can be converted to a general (other than honorable) discharge if you follow the appropriate steps.
You made a mistake. But, it was your mistake, not the Corps' However,, if you learned the core values of Honor, Courage, and Commitment, you will move forward from this incident, get the help you need to address your issues.
You cannot sue the Marine Corps, because of the Feres Doctrine which gives the Marine Corps sovereign immunity from such lawsuits. To challenge your discharge, however, you may submit an application to the Board for Correction of Naval Records (see link below). If you can provide evidence to show that your discharge was unjust, you may be able to be reinstated (if that is what you desire). You should probably see a local military lawyer for help in preparing for BCNR application as there are strict legal requirements you must follow.
You have received some very good information. The Board of Corrections for Naval Records will be your best bet to get your discharge upgraded. Additionally, it is correct that you typically cannot sue the United States Government. However, there are exceptions.
One of the most significant exceptions is the Federal Tort Claims Act (28 U.S.C. § 1346(b)) In essence, this waives the Government's Sovereign Immunity in the case of Liability "circumstances where the United States, if a private person, would be liable to the claimant in accordance with the law of the place where the act or omission occurred."
It is nearly impossible to say whether or not you might have a claim without a lot more information. I would strongly recommend you sit down for a consultation with an attorney who specializes in representing military and former military members.