To give a serious answer: The short answer is probably, the long answer is no.

The opinion was deliberately vague on that issue. A dissent said they could under Roberts' opinion, but Roberts calls that "fear mongering" without elaborating whether that's true or not.

It's also a pretty complicated opinion so bear with me. The whole thing comes down to this vague idea of official vs. unofficial acts which are supposed to be immune according to the court. Really, there's multiple factual allegations and the court said each one has some level of immunity (and if you think these are full of contradictions, I know):

• Asking the DOJ to pressure states to investigate obvious spurious "fraud" claims and pressure states to throw out their results, and threatening to fire them if they refuse - here Trump is "absolutely immune" because the DOJ is part of the executive branch and the president has power to fire them, I guess for any reason now.
• Trying to get Mike Pence to refuse the vote count and throw the whole country into a chaotic power struggle - presumptively immune, because the president and vice president can talk about their duties. Can be rebutted if the government can prove a prosecution wouldn't pose a danger of intrusion into executive branch functions, whatever the hell that means.
• Trump personally telling state officials to change electoral votes - here Roberts says there's no basis for Trump to claim immunity because there's no presidential power to try and coerce state officials. However, he then says it's up to the lower court to consider if it's official or not before proceeding, and is entirely unclear on who has the burden of proof here.
• Using twitter and a speech to organize and then start a riot at the capitol - similar to the above, the president has official duties relating to speaking but yada yada yada it's sent back to the lower court to decide whether this is official or not.

Conclusion: Ordering an assassination of a rival certainly sounds most like the first - the president has several official duties relating to giving military orders, and the military is part of the executive branch. The FBI is also part of the DOJ, so if Trump can order the DOJ to do something criminal, that itself could be an assassination. But as described in the article below, one could make an argument that no, the opinion doesn't actually say he do that with the military specifically, because congress has some powers relating to war (not convincing). However, to be fair to that opinion, this immunity ruling is such a stinker that lower and future courts will limit its holding as much as humanly possible. Plus seemingly contradictory aspects to it (Trump can order the DOJ to do things he can't do himself?) could be used to argue for exceptions to the overall immunity. But reading the opinion at face value, yes the president could order an assassination, and even fire generals who refuse to pass along those orders.

Longer answer though: This is the real world. If Biden gave such an order, it would likely result in a coup and an overthowing of the Constitutional order as a whole. And if order were somehow restored and Biden brought up on criminal charges, you could be your life that the 6-3 Republican majority on the court would find a way to either limit or perhaps overturn their prior ruling as it pertains to Biden.

For an alternative perspective on the same topic, here's a center-to-slightly-right-leaning law professor's take on this which does a pretty plausible job sane-washing the opinion.