The Constitution does not state what the president may or may not do with regard to the writ.
Jiminy Cricket, Andrew Jackson suspended the Writ and he wasn't even president!
"After the battle of New Orleans, and while the fact that the treaty of peace had been concluded, was well known in the city, but before official knowledge had arrived, Gen. Jackson still maintained martial or military law. Now, that it could be said the war was over, the clamor against martial law,, which had existed from the first, grew more furious. Among other things, a Mr. Louiallier published a denunciatory newspaper article. Gen. Jackson arrested him. A lawyer by the name of Morel procured the United States Judge Hall to issue a writ of hebeus corpus to release Loualier. Gen. Jackson arrested both the lawyer and the judge. A Mr. Holander ventured to say of some part of the matter that "it was a dirty trick." Gen. Jackson arrested him. When the officer undertook to serve the writ Gen. Jackson took it from him, and sent him away with a copy.
Holding the judge in custody for a few days, the general sent him beyond the limits of his encampment, and set him at liberty with an order to remain till the ratification of peace should regularly be announced, or until the British should have left the coast. A day or two elapsed, the ratification of a treaty of peace was regularly announced and the judge and the others were fully liberated. A few days more and the judge called Gen. Jackson into court and fined him $1,000. The general paid the fine, and there the matter rested for nearly thirty years, when Congress refunded principal and interest. The late Senator Douglas then in the House of Representatives, took a leading part in the debates, in which the constitutional question was much discussed. I am not prepared to say whom the journals would show to have voted for the measure.
It may be remarked: First, that we had the same Constitution then as now; secondly, that we then had a case of invasion, and now a case of rebellion; and thirdly, that the permanenet right of of the people to Public Discussion, the liberty of speech and the Press, the trial by jury, the law of evidence, and the Habeus Corpus, suffered no detriment whatever by that conduct of Gen. Jackson, or its subsequent approval by the American Congress."
-- A. Lincoln, 6/12/63
Walt
Actually it had been assumed by representatives of his government. The first political prisoner of the war was some poor shmuck of a newpaper reporter who Braxton Bragg tossed into jail on April 13, 1861 for printing an article on southern military preparations.
Another major diffrence was that people arrested in the North by military officials or impacted by actions of the government did not lose their rights of appeal through the federal court system. In some cases, Walt's beloved Prize Cases for example, the court ruled for the government. In other cases, Ex Parte Milligan for another example, the courts ruled against the government. People arrested by the Davis regime had no such avenue of appeal. Once locked up they stayed locked up unless the government let them out because there was no judicial system for them to take their case to. No courts to act as a control on the excesses of the government. No protections of any kind.