ERJ-170 flight crew reported a near miss with another aircraft taking off on a crossing runway while on approach.

Narrative

We were on a normal ILS approach in VMC conditions for Runway XX; and started hand flying at approximately 1000' AGL. At 500' AGL my copilot called out 500 Stable" as normal; immediately after hearing this I heard ATC clear another aircraft for takeoff on Runway XY. This was very close in to the landing for us to have to deal with a potential crossing traffic for our runway. I immediately looked over to that runways and saw the aircraft was slowly accelerating on their takeoff. When at 300' AGL or so; I noticed the taking off traffic was around 1500' down Runway XY and slowly accelerating and knew that we would meet at the runway intersection if we continued the approach and landing. I called for and initiated the go around. As we were climbing through 400' ATC told us to fly a heading and climb to 2000'. My copilot then proceeded to set the guidance panel for this while I flew the aircraft following ATC instructions. At around 500' we got a Traffic alert via TCAS but no RA due to the taking off traffic on XY as we passed above them. As we were dealing with this and flying ATC asked numerous times for the reason for our go around. We finally had to tell them to just stand by on this request. At 1100-1200' ATC gave us another heading and continued climb to 3000'; and again asked for our reason for the go around; then switched us to Departure Control for radar vectors back to ZZZ. Once level at 3000' we asked for delaying vectors while we got through all our checklists. This was accepted by ATC and they again asked for the reason for the go around. I then told them it was due to conflicting traffic on takeoff crossing our runway and we didn't feel safe continuing the landing. We then finished all of our checklists and proceeded to do another approach and landing. When switched to Tower; we AGAIN were asked for our reason for go around. I again told them there was not enough room for the crossing traffic taking off on Runway XY. The controller then attempted to argue and stated "Oh there was plenty of room". As we passed with 300-500' of the traffic taking off; there is no doubt in me that we would have had either a collision or a wake turbulence issue at the intersection of the runways during out first landing attempt."