I'm rather sure the UK would object (and the US would take that into account) because of the Falklands war and Argentina not having given up on that territorial claim. Interestingly, a quick search finds that Argentina may have gotten F-16s at one point, so the US objections weren't super strong.
Later on, in 1998, Argentina would be declared a ‘Major non-NATO Ally’, the sixth nation in the world to receive the title after Australia, Egypt, Israel, Japan, South Korea, and New Zealand. During this time period, The United States made an offer to sell 36+6 F-16A/Bs, unfortunately, due to financial insecurities, the Argentines declined the offer.
After the UK had rejected Argentina being able to access dozens of
aircraft due to their British-built Martin-Baker Ejection Seats, there were only a handful of options left for Argentina.
UK-Argentina relations seems to have worsened a bit lately.
More recently, the US seems to have blocked some sales too, even of the F-16 kind.
The Mirages left service in mid-2015. To rebuild its supersonic fighter capability, Buenos Aires considered buying new Gripens from Brazil, old American F-16s, second-hand Mirage F.1s from Spain and new FC-1s from China before finally settling on 14 used Israeli Kfir Block 60s, priced to move at $350 million for the lot, including several years of maintenance. [...]
In any event, the United States refused to grant an export license for the Kfir’s American-made engines.
I'm not 100% sure that was the reason; some Israeli sources leave that out and invoke more generic political climate cancellation reasons. There's more than one US source claiming that, but they all use similar wording, so there may be some churnalism at play there.
OTOH, other US and Argentine sources claim that the US was offering some old F-16s again in 2020. Anyhow, the F-35 are modern, so both more expensive [Argentina still has troubled finances IIRC] and [much] less likely to get approvals from the West.