In 2013, the United States Supreme Court heard a landmark case in Missouri v. McNeely. In McNeely, the Supreme Court held that under typical circumstances, the Fourth Amendment prevented police from taking a blood sample without first obtaining a warrant.
Prior to that case, the state of Missouri had argued that it could require people accused of driving while intoxicated to give blood samples without obtaining a warrant. This decision was a major victory for defendants and requires that officers obtain a warrant from a judicial authority before engaging in a blood draw, which the Supreme Court noted is an invasive procedure.
In contrast, the United States Supreme Court has held that states can punish individuals who refuse to take a breath test, even if there is no warrant, in Birchfield v. North Dakota, since the breath test is less intrusive.