No. The Republicans were a third party, at a time when the Whigs had a record of electing presidents and controlling Congress. This is not to say that there are many exceptions to the assertation he makes, though.
It could be argued that Millard Fillmore was actually the Whig candidate in 1856, the first year the Republicans fielded a candidate, explorer John C. Fremont. James Buchanan won a three-way contest that year with 45% to Fremont's 33% and Fillmore's 22%. But Fillmore (even as a former Whig President) did not use the Whig Party line. He went by the American Party. The Whig's did, of course, run a few more people for office on the state and local level, but the last national election where they had any role was the contentious election of 1860. Their candidate, John Bell, ran under the Constitutional Union banner and was third runner-up behind Lincoln (39.7%), Douglas (29.5%) and Breckenridge (18.2%) though he actually managed to win three states (Virginia, Tennessee and Kentucky) to one for Douglas (Missouri).
Interestingly, Lincoln was a former Whig himself and changed the name of the Republican Party to National Union Party, selecting the only Confederate U.S. Senator to remain loyal to the Union (and former Whig) Andrew Johnson as his running mate.