Arkansas officials have said they are unable to obtain the necessary drug from any other source, and have acknowledged in court papers that should McKesson prevail, all pending executions would be effectively blocked.
Arkansas plans to execute three more inmates by the end of the month, when its supply of midazolam expires.
Lee was pronounced dead at 11.56pm - 11 minutes after the three drug lethal injection cocktail began to pump into his veins.
Lee was put on death row for the 1993 murder of his neighbour Debra Reese, whom he struck 36 times with a tyre tool her husband had given her for protection. Lee was arrested less than an hour after the killing after spending some of the 300 dollars he had stolen from Mrs Reese.
Two executions are set for Thursday, followed by another double execution Monday and a single execution April 27.
The state originally set eight executions over an 11-day period in April, which would have been the most by a state in such a compressed period since the U.S. Supreme Court reinstated the death penalty in 1976. But the Arkansas supreme court vacated Griffen's ruling days after he participated in an anti-death penalty rally and reassigned some of his cases.
An inmate set to die Thursday night is asking the Arkansas Supreme Court to block his execution so he can pursue more DNA tests in hopes of proving his innocence.
The state high court decision was one of two setbacks Wednesday to Arkansas' bid to resume capital punishment after a 12-year hiatus.
With his arms extended, covered with a sheet, and his head and hands covered with leather straps, Lee made no final statement and showed no apparent signs of suffering during the execution. A protester and a spokesperson for the governor reacted to the controversial execution. "Justice was carried out".
The lawyers have also fought the state's decision to conduct the series of executions in a short time frame, which the state has said is because its lethal-injection drugs will expire at the end of the month.
38 detained in Turkey for questioning legitimacy of referendum result
US President Donald Trump called Erdogan to congratulate him on his victory, while European Union leaders gave more reserved responses.
The court denied an appeal from Ledell Lee, the other inmate scheduled for execution on Thursday - but his execution, along with all the others on the calendar, is still blocked by a second court ruling.
"I pray this lawful execution helps bring closure for the Reese family", Arkansas Attorney General Leslie Rutledge said in a statement. Midazolam, along with the paralytic vecuronium bromide, and potassium chloride which in heavy doses will stop the heart, comprise the state's three-drug lethal injection procedure.
McKesson said it wants nothing to do with executions and was disappointed in the court's ruling.
Hikma, the British manufacturers of the drug, have opposed its use in executions, and the makers of other drugs have tried to halt the executions in the courts, arguing that the prison authorities were unauthorised to use their products.
McKesson Corp says it sold the drug vecuronium bromide to the Arkansas department of correction for inmate medical care, not executions.
Thirty-one states now administer the death penalty, and lethal injection is the primary means of execution in all of them.
"Through the manipulation of the judicial system, these men continue to torment the victims' families in seeking, by any means, to avoid their just punishment", prosecutors said in a joint statement issued Thursday. Attorneys for the inmate filed a request Wednesday for a stay with the state's highest court.
In 2015, the Supreme Court ruled that states could use midazolam, but that has not stopped its use from tangling up executions in courts or leaving states struggling to obtain the drug. At one point in the proceedings before a federal judge last week, Arkansas Solicitor General Lee Rudofsky declared, "Enough is enough".
But the US Supreme Court in a 5-4 ruling reversed the lower court's decision on Thursday night.
Cut off from the supply of drugs, many states had to put off executions or experiment with new lethal injection combinations. "In my view, that factor, when considered as a determining factor separating those who live from those who die, is close to random", Breyer wrote.