Mysterious RX pulses on UART connect on OS X Arduino Due
Short:

Looking at the ATMEGA16U2 firmware (https://github.com/arduino/ArduinoCore-sam/blob/master/firmwares/atmega16u2/arduino-usbserial/Arduino-usbserial.c) I find that, when you configure/change settings of the USB emulated serial port, the USART is resetted. This happens even when you open the Arduino Serial Monitor (it must configure the serial speed, etc.). This causes your spike.

Long:

Look at function:

void EVENT_CDC_Device_LineEncodingChanged(USB_ClassInfo_CDC_Device_t* const CDCInterfaceInfo) 

There you'll see that after some lines, it resets the USART, by zeroing its registers:

/* Must turn off USART before reconfiguring it, otherwise incorrect operation may occur */
    UCSR1B = 0;
    UCSR1A = 0;
    UCSR1C = 0;


At page 168, of the current ATMEGA16U2 datasheet, you'll find that, by setting  UCSR1B's bit 3 (TXEN1), you enable the transmitter,  overriding the normal port operation (i.e. it becomes output). Quoting the datasheet:


  Writing this bit to one enables the USART Transmitter. The Transmitter
  will override normal port operation for the TxDn pin when enabled. The
  disabling of the Transmitter (writing TXENn to zero) will not become
  effective until ongoing and pending transmissions are completed, i.e.,
  when the Transmit Shift Register and Transmit Buffer Register do not
  contain data to be transmitted. When disabled, the Transmitter will no
  longer override the TxDn port.


Therefore, by writing UCSR1B = 0; you no longer override the TXD1 pin, which will act as input.

The ATMEGA16U2 TXD is connected to the ATSAM3X8E's RX line. In normal operation, with the UART enabled, that line stays high if no data is being transmitted. If you disable the UART, that particular line is no more driver to 1. Since the initialization code does not set the pull-up on that pin (and neither it's configured as output), the pin becomes a floating input, and any leakage to GND or even the input impedance of your probe (which is between your pin and GND), will slowly bring the logic level to 0.

To override this, problem, you should either:
1) Modify ATMEGA16U2 firmware, by setting that PIN as OUTPUT, with value 1.
2) Modify ATMEGA16U2 firmware, by enabling the pull-up on that pin.
3) (suggested) Enable the pull-up on the RX line on the ATSAM3X8E.