There exists no gemstone designated as a traditional gift for the 60th or any birthday. Gemstones associated with births are designated based upon the month a particular individual is born, not the person's age.
However, wedding anniversaries are celebrated with traditional gemstones as gifts. The traditional gemstone associated with the 60th wedding anniversary is the diamond. The birthstone for January is garnet, for February is amethyst and for March is aquamarine; April is celebrated with the diamond, May with the green emerald and June with the pearl; the ruby is the gemstone for July birthdays, peridot for August birthdays and sapphire for September. Finally, October birthdays are marked with the opal, November with yellow topaz and December with blue topaz.