By the fact that not giving consent requires action, not inaction.
But that's not a fact. If you ask for someone's consent, and they don't respond, they haven't given consent. Inaction, by definition, means that consent was not "given". The only time the law implies that "silence is consent" is when it is expressly stated as such. And the Constitution does not say that.
The constitution dictates that the President "shall nominate and with the Advice and Consent of the Senate, shall appoint... Judges of the Supreme Court", so the Senate needs to either give consent or not give consent. Either result requires Senate action, you argue that inaction is perfectly fine.
You are interpreting the word "consent" as if they had made a requirement that the Senate "object".
If the Constitution stated that "the President's nomination shall be considered appointed unless a majority of the Senate objects", such a provision would require Senate action. But that's not what it says. Consent requires an affirmative act, and the Senate need not consent if it does not want to. Legally, a Senate non-vote, and a Senate vote against consent, mean the exact same thing.
Your emphasis on the word "shall" doesn't make sense, because it doesn't say that the Senate "shall" vote. That second "shall" simply means that if the President nominates a justice, and if the Senate consents, the President "shall appoint" the nominee to the Court. It's actually a
three step process - nomination, confirmation/consent, appointment.
Your application of the "shall" requirement to the
Senate action makes no sense, because it would mean that the Senate
shall (i.e., "must) "consent", which would be nonsensical. It would mean that the Senate must confirm all of the President's nominees.