If [he was compelled to put his finger on the phone], then that's incontrovertibly a violation of the Fifth Amendment.
As someone who used to stand by that view, nowadays it strikes me as the stance of someone who values their privacy (as we all should!), but who hasn't thought through the ramifications of their stance yet.
For instance, I'd wager you have no problem when the police swab a suspect for their DNA, nor when a passed-out drunkard is compelled to provide a blood sample in the hospital after a DUI, yet in both cases the suspect is being compelled, potentially against their will, to provide something incriminating of themselves to a machine in the police's custody that will tell the police whether the evidence from the suspect is incriminating or not. That's no different than compelling a suspect to provide their fingerprint to a phone in the police's custody that may have the ability to incriminate the suspect.
In fact, both DNA evidence and the BAC measurement situation I described have made it through and been affirmed by the Supreme Court already (in some cases, multiple times), for the simple reason that the right against self-incrimination only extends to "testimonial" evidence (a.k.a. "communicative" evidence), not to "real" evidence...nor should it.
I recall reading portions of the majority opinions for some of the seminal cases in this area a year or two back when researching the topic, and one of them basically stated that if we took the notion that we can't collect incriminating "real" evidence to its logical conclusion, we wouldn't even be able to compel someone to reveal enough of their physical appearance for them to be recognizable to an eyewitness, which they asserted was utterly absurd and was clearly beyond the bounds of the protections afforded by the 5th Amendment. More or less, so long as the police have a warrant and aren't trying to compel any form of demonstration of knowledge (i.e. testimony), they're within their rights.
You've already said that you're fine with the police collecting fingerprints, which is good, since fingerprints are not testimonial/communicative in nature. But how the police collect and use them is left up to them to decide. Whether they collect them on a piece of paper, via an electronic scanner that stores them to local database, or by way of a sensor that writes them into a transient piece of memory on a mobile device makes no difference. In all three, they're simply compelling the suspect to provide a piece of evidence in their custody to a device or system in the police's custody. It's a simple transfer of physical evidence from the suspect to the police. The means may be different, but the thing being compelled is the same in all three cases.
That the evidence can be used to incriminate the suspect does not mean their rights have been violated. And the best course of action if you don't like that fact is to stop using real evidence (e.g. keys, fingerprints, etc.) as a locking mechanism.