This quote is correct with regard to presidential elections - the US presidential election has never been won by a candidate which was voted for by the majority of the voting-eligible population. Below is a chart created using data on VEP turnout from here, and overall presidential election result data from here. It shows that the largest proportion of the VEP that has ever voted for the winning candidate was 42.6%, for Ulysses Grant in 1868. At that election, the percentage of the VEP which voted was 80.9%, and of those voters, 52.66% voted for Grant.

Year,VEP Turnout Rate,% of Popular Vote,Winner,% of VEP voting for winner
1789,11.6%,100.00%,George Washington,11.60%
1792,6.3%,100.00%,George Washington,6.30%
1796,20.1%,53.45%,John Adams,10.74%
1800,32.3%,61.43%,Thomas Jefferson,19.84%
1804,23.8%,72.79%,Thomas Jefferson,17.32%
1808,36.8%,64.73%,James Madison,23.82%
1812,40.4%,50.37%,James Madison,20.35%
1816,16.9%,68.16%,James Monroe,11.52%
1820,10.1%,80.61%,James Monroe,8.14%
1824,26.9%,30.92%,John Quincy Adams,8.32%
1828,57.3%,55.93%,Andrew Jackson,32.05%
1832,57.0%,54.74%,Andrew Jackson,31.20%
1836,56.5%,50.79%,Martin Van Buren,28.70%
1840,80.3%,52.87%,William Henry Harrison,42.45%
1844,79.2%,49.54%,James Polk,39.24%
1848,72.8%,47.28%,Zachary Taylor,34.42%
1852,69.5%,50.83%,Franklin Pierce,35.33%
1856,79.4%,45.29%,James Buchanan,35.96%
1860,81.8%,39.65%,Abraham Lincoln,32.43%
1864,76.3%,55.03%,Abraham Lincoln,41.99%
1868,80.9%,52.66%,Ulysses Grant,42.60%
1872,72.1%,55.58%,Ulysses Grant,40.07%
1876,82.6%,47.92%,Rutherford Hayes,39.58%
1880,80.5%,48.31%,James Garfield,38.89%
1884,78.2%,48.85%,Grover Cleveland,38.20%
1888,80.5%,47.80%,Benjamin Harrison,38.48%
1892,75.8%,46.02%,Grover Cleveland,34.88%
1896,79.6%,51.02%,William McKinley,40.61%
1900,73.7%,51.64%,William McKinley,38.06%
1904,65.5%,56.42%,Theodore Roosevelt,36.96%
1908,65.7%,51.57%,William Taft,33.88%
1912,59.0%,41.84%,Woodrow Wilson,24.69%
1916,61.8%,49.24%,Woodrow Wilson,30.43%
1920,49.2%,60.32%,Warren Harding,29.68%
1924,48.9%,54.04%,Calvin Coolidge,26.43%
1928,56.9%,58.21%,Herbert Hoover,33.12%
1932,56.9%,57.41%,Franklin Roosevelt,32.67%
1936,61.0%,60.80%,Franklin Roosevelt,37.09%
1940,62.4%,54.74%,Franklin Roosevelt,34.16%
1944,55.9%,53.39%,Franklin Roosevelt,29.85%
1948,52.2%,49.55%,Harry Truman,25.87%
1952,62.3%,55.18%,Dwight Eisenhower,34.38%
1956,60.2%,57.37%,Dwight Eisenhower,34.54%
1960,63.8%,49.72%,John Kennedy,31.72%
1964,62.8%,61.05%,Lyndon Johnson,38.34%
1968,62.5%,43.42%,Richard Nixon,27.14%
1972,56.2%,60.67%,Richard Nixon,34.10%
1976,54.8%,50.08%,Jimmy Carter,27.44%
1980,54.2%,50.75%,Ronald Reagan,27.51%
1984,55.2%,58.77%,Ronald Reagan,32.44%
1988,52.8%,53.37%,George H. W. Bush,28.18%
1992,58.1%,43.01%,Bill Clinton,24.99%
1996,51.7%,49.23%,Bill Clinton,25.45%
2000,54.2%,47.87%,George W. Bush,25.95%
2004,60.1%,50.73%,George W. Bush,30.49%
2008,61.6%,52.93%,Barack Obama,32.60%
2012,58.6%,51.06%,Barack Obama,29.92%
2016,60.1%,46.09%,Donald Trump,27.70%