The phrase "asserted he would" is correct and usable in written English.
It can be used when reporting someone's statement or claim about their intentions or future actions.
Example: "During the meeting, he asserted he would complete the project by the deadline."
Alternatives: "claimed he would" or "stated he would".
Mr. McCain asserted he would do well in several states with closed primaries, particularly Connecticut, Massachusetts and Rhode Island.
ORLANDO – By the fifth and final stop of his Florida Fly-Around tour, Mr. Gingrich asserted he would win Tuesday's crucial primary in this all-important state.
ISLAMABAD, Pakistan — President Pervez Musharraf fought back Friday against politicians who asserted he would resign rather than face impeachment charges, saying through his chief political supporter that he would challenge the charges when they were brought to Parliament.
Prime Minister Tony Blair said on Tuesday that he had no question that his decision to go to war in Iraq was right, and he asserted he would do it again in the same circumstances.
Earlier this year Prime Minister David Cameron made comments on the tech widely interpreted as an intention by a Conservative-led government to seek to ban encryption — after he asserted he would not allow a means of communication that "even in extremis, with a signed warrant from the Home Secretary personally… cannot [be] read".
However, the 64-year-old asserted he would still have helped the club challenge for promotion to the Premier League - and believes they could still go up under his replacement, Nigel Pearson, even though they are seven points adrift of the play-off places.
Cruz, in a victory speech after his win in Wisconsin this month, asserted he'd clinch the nomination "either before Cleveland, or at the convention in Cleveland".
Had he lived a century later, Stiles asserts, "he would have been called a terrorist".
Bezos called Ashley in response, asserting he would fix the problems and had Ashley formally reinstated as an Amazon employee.
If Nietzsche's image reached its nadir during the Second World War, when Hitler presented Mussolini with a bound edition of his works and the historian Crane Brinton wrote a book asserting he would have been "a good Nazi," a resurrection was soon to come.
We made national news when I quoted him asserting he would not try to capitalize on his new-found prominence by running for president in 1976, though he did try to mount a presidential bid in 1980.
