The accepted answer shows the legal reason. But this is politics.SE, so I'll provide some examples that demonstrate the political nightmare this would cause.
The president cannot make a law to protect only himself
Even if he could technically, it would be political suicide to make a law that says "Thou shalt not kick the president in the knee. This law applies only to the president." It would have to apply to everyone.
This could backfire and cause problems for people reporting the truth
Let's borrow Philipp's example for a moment. Let us suppose that Philipp actually did kick his dog. You report this since it is true: "I just saw Philipp kick his dog." If Philipp can show that the claim is unfounded, or worse, lie about it and show (falsely) that you are making a false accusation against him, then you will get charged for your "false" reporting and get penalized for telling the truth.
The reporter might believe that what they are reporting is true
Let's twist the previous example a bit. What if it was someone in a Philipp costume who kicked Philipp's dog, and they did it to fool you. You have now been fed false news, but you have every reason to believe it is true. You report on what you reasonably believe to be true, but Philipp later shows it to be false.
How does your proposed law handle mutually exclusive beliefs by opposing points of view?
A Christian, a Muslim, and an Atheist walk into a news station. No, this is not the setup for a joke, though each one of them might believe the other to be a laughingstock.
The Christian reports:
Jesus arose from the dead! After being sacrificed for our sins, he has
conquered death. All who believe on him and repent can be saved from
The Muslim reports:
Jesus was a fraud! His disciples removed his body from the tomb then
claimed he arose and left.
The Christian files a claim against the Muslim using a new law that does just what you requested: allows false news to be punished. The Christian's report claims that the Muslim is spreading false lies that the Christians removed the body of Jesus from the tomb.
At the same time, the Muslim files a report against the Christian for making a false claim that Jesus arose from death, hoping to get the Christian punished.
At court, the Christian uses the fact that there were hundreds of witnesses who saw the resurrected Jesus walking and talking and still having the wounds from his crucifixion. The Muslim says those witnesses don't count since it was a long time ago and all those witnesses are now dead; there is nobody to bear witness in court today.
The courts cannot say the Christian made a false news story. To do otherwise would set a terrible precedent: someone reporting on the crusades or the Spanish inquisition or any other historic events could likewise face punishment.
The courts cannot say the Muslim made a false news story. To do otherwise would set the precedent that nobody is allowed to challenge commonly held beliefs.
So what do we do now!?? But wait, there is more...
While this is going on, still back at the news station...
The Atheist reports:
There was no Jesus, and there is no God.
Immediately upon finishing the report, the atheist files a fake news claim against both the Christian and the Muslim for their reports about Jesus and previous reports they have made about God.
But wait, there's more! The Christian and the Muslim each heard the Atheist's report, so they both file fake news claims against the atheist.
This is getting out of control!!!
And then a Jew, a Hindu, and a Buddhist each file claims against the reports so far and then walk into the news station...