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Multiply the Amps by the Volts and you get Watts i.e. 115 x 1.5 = 175.5  so say a 200W inverter might do.  Assuming that your ice maker contains a motor driven compressor the following will apply.  To actually start a motor, though,  takes quite a lot more power than it will when it is running steadily, so it could need to be quite a lot larger.  It also could depend on how the inverter responds to momentary overload.  If it cuts out immediately the motor may not get going sufficiently and it will stall.  If the inverter waits until it begins to overheat, on overload, it will probably manage to start it OK.  With an oscilloscope  and a small resistor in series with the compressor- motor you should be able to see what current is drawn and for how long, which should give an indication of the size of supply needed.