Is it possible?
Could the US government print crypto of its own?
This partly depends on how you and the government run institutions define cryptocurrency. This kind of idea is sometimes referred to as a Central Bank Digital Currency (CBDC).
US
The US treasury has published a report on the future of money and payments which
reviews the U.S. system of money and payments, including such recent innovations as
instant payments and stablecoins, and a potential U.S. central bank digital currency (CBDC), and
considers the implications of these developments for key public policy goals including
supporting U.S. global financial leadership, advancing financial inclusion and equity, and
minimizing risks
So the possibility is being investigated, but there is no conclusion about whether it is possible to do this in a way that meets US government objectives.
UK
The Bank of England has published some information about the idea of a CBDC
Note that they say
People sometimes describe this possible UK digital currency as ‘digital sterling’ or ‘Britcoin’.
This creates a conceptual link between Bitcoin and a CBDC. However they also say
A CBDC is different from cryptocurrency (also known as cryptoassets). Cryptocurrencies are not issued by a central bank. These privately issued digital currencies include Bitcoin, Ether (Ethereum) and XRP.
(my emphasis)
This makes it clear that, for them, a key attribute of a cryptocurrency is that they are not issued by a government run central bank.
I think their article is well worth reading. However it seems clear that the Bank of England is not going to introduce a CBDC soon, if it all, and their CBDC might be very different from normal notions about cryptocurrencies.
Venezuela
In 2017, Venezuela announced a government backed cryptocurrency named the Petro.
So you could regard this as evidence that government run central banks can introduce cryptocurrencies.
However it is in doubt whether this is a successful model that other countries would want to follow. The Venezuelan government has had to mandate the adoption of the Petro, for example by requiring all petrol (gasoline) stations to accept it in addition to accepting the Bolivar. Recently they introduced a law that could require a taxation of up to 20% on "large" transactions made using competing cryptocurrencies and foreign fiat currencies.
Readers might like to ponder the likelihood of a US president relishing being seen to follow the Venezuelan precedent.
Is it a good idea?
An anonymous sort of Bitcoin backed by the US government seems like a good idea, no?
It is worth remembering that, Bitcoin, the first successful cryptocurrency, was created by someone who opposed the idea that central banks have any control over people's money.
The root problem with conventional currency is all the trust that's required to make it work. The central bank must be trusted not to debase the currency, but the history of fiat currencies is full of breaches of that trust. Banks must be trusted to hold our money and transfer it electronically, but they lend it out in waves of credit bubbles with barely a fraction in reserve. We have to trust them with our privacy, trust them not to let identity thieves drain our accounts. Their massive overhead costs make micropayments impossible.
- Nakamoto, 2009
I believe it is very clear that Bitcoin was created with the intent of removing the need for banks as intermediaries and without any possibility of government run banks having control over the money supply. These ideas are clearly at odds with any straightforward notion of a cryptocurrency minted by a US government agency.
Bitcoin
Obviously, none of the above can apply to Bitcoin as it exists today. A CBDC would be a separate cryptocurrency. No one national government can take control of Bitcoin globally. Any attempt would fork a parochial altcoin.