The administration claimed Pretti was armed and posed a threat, but bystander video footage reportedly contradicts this, showing him holding only a phone.
Governor Hochul emphasized that "no one is above the law."
This incident is the second time a US citizen has died at the hands of immigration officials in Minneapolis this month, following the earlier killing of Renee Good.
Secretary Noem is also facing criticism from conservative commentators, who allege she is deliberately seeking public confrontations and marginalising other officials.