Strange behavior with CA3130EZ op amp in follower configuration
Yes, as you suspect the CA3130EZ is not unity gain stable without compensation capacitor(s). 
On page 3 of the datasheet states that a capacitor (called \$C_c\$) of at least 47pF be placed from pin 1 to pin 8 of the device. \$C_c\$ provides Miller compensation, and reduces bandwidth of the part. Circuits shown with unity gain also typically have a 2kOhm || 0.1uF feedback network, and a 25pF load. See the Bode plot of open loop response on page 14 for details of effects of various values of \$C_c\$ and \$C_L\$.