The pilot reported that he departed in his retractable gear amphibious airplane from a paved runway and made a climbing left turn to the northwest. He leveled off about 1,200 ft mean sea level (msl), departed the populated area, and then descended to 600 ft msl over a wetlands area. About 10 to 12 minutes after departure, the engine surged several times, and subsequently lost power. Unable to maintain altitude, the pilot elected to make a forced landing on the water. Upon touchdown, the airplane immediately nosed over, which resulted in substantial damage to the fuselage. ADS-B data revealed that the airplane flew parallel to an inland waterway at 500 ft msl, before making a left 180° turn to the southeast, while maintaining level flight. At the completion of the turn, the airplane began a gradual descent, while aligned on final approach to the water. A postaccident examination of the airplane revealed that the landing gear was extended. Following recovery, the engine was allowed to dry, the oil was changed, and a test run was conducted with no anomalies noted. The pilot stated that he did not retract the landing gear after departure and subsequently landed with the landing gear extended. The cause of the reported loss of engine power could not be determined; however, it is likely that, had the landing gear been retracted, a normal landing could have been accomplished safely.