Born in Williamsburg, Brooklyn, New York, Carew attended the public schools of Brooklyn and New York City and the College of the City of New York. He was graduated from Columbia College in 1893 and from Columbia University Law School in New York City in 1896. He was admitted to the bar in 1897 and commenced practice in New York City. He served as member of the State assembly in 1904. He served as delegate to all Democratic State conventions from 1912 to 1924. He served as delegate to the Democratic National Conventions in 1912 and 1924.
Carew was elected as a Democrat to the Sixty-third and to the eight succeeding Congresses. He served from March 4, 1913, until his resignation on December 28, 1929, having been appointed a justice of the New York State Supreme Court. He was subsequently elected to the same office in November 1930 for a fourteen-year term, but retired December 31, 1943, due to age limitation. He served as official referee of the New York Supreme Court. He died in Rockville Centre, New York, April 10, 1951. He was interred in Calvary Cemetery, New York.