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A scheme for measuring small intrinsic critical currents Ic in nanoscale devices is described. Changes in Josephson inductance LJ are converted to frequency variations that are recorded via microwave reflection measurements at 700-800 MHz. The critical current is determined from the frequency shift of the reflection magnitude at zero phase bias assuming a sinusoidal current-phase relation. The method is used to study a multiwalled carbon nanotube transistor with Pd/Nb contacts inside a resistive on-chip environment. We observe gate-tunable critical currents up to Ic [?] 8 nA corresponding to LJ > 40 nH. The method presented is also applicable to devices shunted by closed superconducting loops. [ABSTRACT FROM AUTHOR]