The prediction has two parts, and together they do not hold up.

1. “By June 2021, Covid as a pandemic is going to be over [in the U.S.]”
   • In early June 2021 the U.S. was still recording roughly 14,000 new cases and about 400 deaths per day, with tens of thousands of new infections each week and many restrictions and emergency measures still in place. (archive.cdc.gov)
   • Far from being “over,” the U.S. then experienced a large Delta surge starting in the summer of 2021, with cases and hospitalizations rising sharply as Delta became the dominant variant. (statnews.com)
   • This was followed by the massive Omicron wave in winter 2021–22, which produced the most intense phase of U.S. transmission, with daily new cases peaking at over 1.25 million and a seven‑day average near 600,000 cases in January 2022. (pmc.ncbi.nlm.nih.gov)
   These large national waves after June 2021 indicate that Covid was still functioning as a pandemic in the U.S. well beyond the date he specified.

2. “There will always be cases… It’s going to return as a seasonal illness.”
   • The persistence part is correct: as of 2025, experts widely describe Covid as an endemic virus that continues to circulate year‑round in the U.S., causing recurrent waves rather than disappearing. (washingtonpost.com)
   • However, Covid has not clearly settled into a simple, flu‑like winter‑only pattern. Analyses through 2023–25 emphasize that SARS‑CoV‑2 still produces sizable waves in multiple seasons (including summer), and that its transmission has not yet fallen into a stable, predictable seasonal rhythm. (statnews.com)

Because (a) the time‑specific claim that the pandemic phase in the U.S. would be “over by June 2021” is clearly refuted by subsequent large national waves, and (b) the virus’s behavior to date is more accurately described as year‑round endemic with complex surges rather than a straightforward seasonal illness, the overall prediction is best judged wrong.