Solomon was born in Okeechobee, Florida, the son of Eugenia and Daniel Hunt. He was raised by his mother and stepfather, Seymour Solomon, and his family moved to New York when he was 15. He attended school in Florida and Delmar, New York, and graduated from Delmar's Bethlehem Central High School in 1948. He attended Siena College from 1949 to 1950 and St. Lawrence University from 1953 to 1954.

In the 1950s and 1960s Solomon was involved with M. Solomon, the women's clothing Business owned by his stepfather. In the mid-1960s, he became a partner in the Associates of Glens Falls insurance company, as well as a partner in the Anchor Realty real estate brokerage. In addition he was President of Solomon, Veysey, Dixon, Gohn Associates, an investment brokerage. He also served as President of Queensbury's merchants association, and a member of the board of Directors of the Glens Falls chamber of commerce. He was active in the Jaycees and the Boy Scouts, and was also involved in several charitable and civic causes. In addition, his fraternal memberships included the Marine Corps League, Masons, Shriners, and American Legion. His accomplishments and the name recognition he acquired in the Glens Falls-Queensbury area provided him with an entrée into local politics and government.

In 1955 Solomon married Freda Parker of Monongahela, Pennsylvania. They were the parents of five children, Susan, Daniel, Robert, Linda and Jeffrey. In addition to his wife and children, Solomon was survived by six grandchildren and his brother Richard.

Originally a Democrat, from 1968 to 1972 Solomon served as Queensbury's Town Supervisor, which also made him a member of the Warren County Board of Supervisors. In February 1968 he switched his party affiliation to Republican, arguing that he had been a Republican prior to the election of John F. Kennedy, that he joined the Democrats because he was inspired by Kennedy's goals and vision, but that he was returning to the Republicans because he perceived that Lyndon B. Johnson's presidency had moved too far from Kennedy's agenda. He was a member of the New York State Assembly from 1973 to 1978, sitting in the 180th, 181st and 182nd New York State Legislatures.

In November 1978, Solomon was elected to the House of Representatives. In 1980, he was an early and avid supporter of Ronald Reagan's presidential campaign, and remained a Reagan stalwart, supporting his efforts to reduce taxes, revive the economy, halt Soviet expansionism and rebuild America's defenses. Solomon supported the line-item veto, a constitutional amendment to ban flag burning, a repeal of the Federal Assault Weapons Ban, and amendments that strengthened the Selective Service System. He also sponsored the legislation that created the United States Department of Veterans Affairs, granting full cabinet status to the former Veterans Administration. He was such an opponent of gun control that he once challenged fellow Representative Patrick J. Kennedy of Rhode Island to "step outside" to settle a dispute over the issue. Solomon attempted to run for Speaker of the House in 1994 but withdrew in favor of Newt Gingrich.

Solomon was named ranking Republican of the House Rules Committee in 1991. Even though Jimmy Quillen of Tennessee had far more seniority, the other Republicans on the committee thought Solomon was a more accomplished legislator. When he lost his bid for the Speakership in 1995, he became chairman of the Rules Committee, thus making him one of the three or four most powerful members of the House. He served as chairman from 1995 to 1999, when he retired.

After three terms in the Assembly, Solomon won election to the U.S. House. He became a high-profile advocate on defense and veterans' issues, and was best known for passage of his Solomon Amendment, which prohibited federal funding to colleges and universities that banned military recruiters from their campuses. After Republicans won control of the House in the 1994 elections, Solomon challenged Newt Gingrich for Speaker, but withdrew when it became clear that Gingrich had the support of the Republican caucus. Solomon became chairman of the Rules Committee, a post he held from 1995 until retiring from the House in 1999.

On October 26, 2001, Solomon died of congestive heart failure at his home in Queensbury at the age of 71. His ability to maintain friendships across political lines was demonstrated when one of the eulogies at his funeral was delivered by Representative Charles Rangel, a Democrat and fellow veteran. Solomon was buried at the national cemetery which is named for him.

Solomon was also the originator and sponsor of the Solomon Amendment, a controversial amendment to United States Code that precluded the receipt of federal government funds by colleges and universities unless they provided equal access to military recruiters as they did private employers. The policy was a response to rules at many American law schools which excluded employers who discriminated based on categories such as sexual orientation. The Amendment was subject to extensive litigation in 2003 by plaintiffs; the Forum for Academic and Institutional Rights (FAIR) and the Society of American Law Teachers. See FAIR v. Rumsfeld, 390 F.3d 219 (3rd Cir. 2004). On March 6, 2006, the United States Supreme Court ruled in favor of the military and upheld the Solomon Amendment.