No, absolutely not. A prosecutor (trial counsel) cannot seek reconsideration or appeal a finding of not guilty (an acquittal) by a court-martial panel. The principle of double jeopardy, guaranteed by the Fifth Amendment to the U.S. Constitution and codified in the UCMJ, attaches the moment a trial begins. An acquittal is a final and absolute verdict that the government has failed to prove its case beyond a reasonable doubt. It is a complete bar to any further prosecution for that same offense.
This protection is one of the most fundamental principles of the American legal system, both civilian and military. The government gets one chance to prove its case against an individual. If they fail, that is the end of the matter. There is no process for a prosecutor to ask the panel to “reconsider” their not-guilty verdict or to appeal the acquittal to a higher court, arguing that the panel made a mistake. The panel’s verdict of acquittal is final.
A military defense attorney’s goal is to secure this final and binding acquittal for their client. Once the president of the court-martial panel announces the words “not guilty,” the case is over, and the accused service member is free from that charge forever. The prosecutor must respect this verdict, and any attempt to re-litigate the issue would be a gross violation of the law and the Constitution.