Arrest isn't conviction, but since the topic here is DNA it's worth noting that in California you can have your DNA taken against your will and put into a database when you get arrested even if you're never convicted or even charged with a crime.
https://news.ycombinator.com/item?id=6811167#6811805
Just as there was a state proposition (sponsored by the relatives of one of this killer's victims) to expand DNA-testing of criminal suspects, a future policy change could send investigators into those filing cabinets, if the blood DNA isn't too degraded, to do a broad genetic dragnet for criminal suspects (or their relatives).
Much as you can opt out of the Second Amendment protection of firearm ownership by moving out of the US.
But that's so discouraged, and the normal process is so automatic and hectic, that it's very rare. Few even realize the blood-spot permanent-collection is happening.
There was a proposed bill in 2015 to require signed consent before the collection, but it was defeated. See very end of:
https://nakedsecurity.sophos.com/2015/11/12/california-colle...
Since proposition 69 (2004), and recently upheld by the Calfornia Supreme Court, California is taking and holding indefinitely DNA for everyone arrested for a felony, even if they're never charged or convicted:
http://www.latimes.com/local/lanow/la-me-ln-dna-supreme-cour...
So they get already get 'free look' (no warrant required) at the DNA of anyone alive, via an arrest, even if the charges are ultimately unsupportable. Do you expect a stronger protection would apply against stored records?
Further, the newborn blood spots are already in government custody. There's no essential requirement for any private 3rd party to produce them, against the holder's wishes. Why wouldn't a detective/prosecutor, perhaps emboldened by public opinion or some new Prop-69-like policy, to consider these blood spots 'abandoned' DNA, just like that acquired from this 'Golden State Killer' suspect after surveillance but without a specific warrant?
Is it unlikely that law enforcement would ever get a 'warrant to modern-sequence-all-the-spots', to solve some particularly heinous crime, then retain the data indefinitely, as they already do for other incidental collections? Sure, that sounds to me like a 'general warrant' that should be prohibited by the 4th amendment. But government keeps seeking – and often winning in court! – ever-broader database warrants. (See for example https://www.washingtonpost.com/news/monkey-cage/wp/2016/06/1...)
Even the ~80 yes/no genetic disorders automatically tested for, and thus possibly kept in a digital database afterwards, might be enough 'bits' to narrowly pick a few suspects, or uniquely identify a single suspect, in some cases. Would the CA DOJ need a warrant to have a friend at the CA DPH run a SQL query on a database that may already be online?
Can you find any statement from the California Department of Publish Health that they would only release blood spot data to law enforcement with a warrant?