George began his career with his local junior side Irvine Meadow before signing for Queen of the South during the 1937 close season. At Dumfries, George was spotted by Billy Halliday, the brother of Dons manager Dave, and in the closing weeks of the 1937-38 season Aberdeen paid out a then club record of £3,000 for his services. Almost immediately "Hammy" proved his worth with exciting displays of his talent in the 1938 Empire Exhibition Tournament. George was principally an inside forward who had great natural inability, was happy with the ball on either foot and had an obvious love of playing the game. He was a good header of the ball and was also a more than adequate leader of the line if called on to play in the centre forward's role. The outbreak of the Second World War deprived George of what should have been the best days of his footballing life but he returned to Pittodrie after the war aged 27, to write his name in large letters in the club's history books.

George was hugely influential in the Dons 1945-46 Southern League Cup triumph and a year later scored the equaliser as the Dons won the Scottish Cup for the first time in their history as they came back to beat Hibs 2-1.

The Scottish selectors were not blind to Hamilton's talents and in November 1946 he received a call up to play for the national side on that Hampden international versus Northern Ireland. George acquitted himself well in the game but curiously he became the forgotten man of Scottish international football and despite continuing to sparkle on a weekly basis with the Dons (and Hearts for a six month spell before returning again to Pittodrie), he was not put back in to the international frame again until May 1951 when he was selected for a game in Brussels against Belgium. On that occasion Hamilton lined up at centre forward and he put on a superb display to claim a hat-trick in a 5-0 win.

The following week George lined up at centre again in an ill tempered game in Vienna against the Austrians but the side went down 4 - 0 and George Hamilton's Scotland days appeared to be over. Incredibly, in May 1954 as Scotland prepared for the 1954 World Cup finals in Switzerland, 36 year old George was recalled to the Scotland side to face Norway. George did not let the selectors down and scored the only goal of the game from his natural inside right position. He faced the Norwegians again in Oslo a fortnight later but alas there was to be no World Cup finals postscript to George's international career. A year later after only a handful of cameo roles in Aberdeen's triumphant 1954-55 championship season George was allowed to leave Pittodrie for Hamilton Accies but after only six months at Douglas Park George finally decided to hang up his boots.