Calibration disc? How is that going to do anything useful for internet content? Presumably, if you are doing Blu-ray you already HAVE a test/setup disc anyway and the settings you picked for Blu-ray will be the same settings you use for streamed content. Those discs are hardly "calibration" discs as all they do is help you find the best settings for Brightness/black level and sharpness. Everything else they (claim to) do is subject to such large errors that you can't call it "calibration" by any stretch. And you can't even trust the results you get for setting color and tint because the instructions always say that if the results don't look right, tune color and tint by eye until they look right.
Then someone else said "all devices are different" - that used to be true for analog video. In digital video, each device in the chain is passing 1s and 0s. No device in the chain should change the 1s and 0s unless there is a control you set that intentionally alters them. If you view Dawn of the Dead on Blu-ray or via internet streaming, both will have originated from the same digital master (presumably). The difference will be that even at the highest streaming speeds, the video has to be compressed more than it is on Blu-ray. Compression won't alter color or grayscale. Transmitting the compressed video over the internet won't alter color of grayscale. Your only worry is that something in the path from the master to the compressed video DOES mangle the original video in some way... if it does, it's really crap equipment. All consumer HD video has the SAME standard... Rec. 709. So whatever the right settings are for Blu-ray should be the right settings for a cable/satellite box or internet streaming. Some product along the way that randomly alters 1s and 0s will cause complete loss of image. If the product "intelligently" alters the 1s and 0s it might change color or luminance or some other parameter, but mostly, the only alteration of 1s and 0s will be to compress the image and that won't alter color or grayscale unless it is extraordinarily crappy compression. Once the compression is done, the decompression for display will retain the original color and grayscale but you'll find more compression artifacts in the image.
Streamed internet content is a lot like 300 channels of cable or satellite TV. They all "know" the "target" is the Rec.709 HDTV color standard. If you think of all those channels and streamed movies as pellets in a shotgun shell and you shoot that shell at a big archery target, many of the pellets will hit the bullseye, but some are going to "spray" other areas of the target without hitting the bullseye. Since the majority of programs "hit the target" with some relative accuracy, any changes you make that move the video display away from the Rec. 709 HDTV video standard is going to make things LESS accurate on average. So all you can do to "calibrate" for internet content or for cable/satellite content is to REALLY calibrate the video display to get as close to the Rec. 709 HDTV video standard as it can be. That requires a colorimeter, calibration software, a test pattern source (could be a disc in a disc player), and knowledge about calibration... that can be a DIY thing or you can hire a professional calibrator.
If some source/provider doesn't adhere to the Rec. 709 HDTV specification, there's not much you can do about it. But understand... once the video is compressed, it is VERY difficult to change the images... EXTREMELY difficult in fact. Because any "accidental" changes to the bitstream would likely result in complete loss of the image... you just can't randomly diddle with a compressed digital video stream and affect color or grayscale because the bits are in a very controlled pattern that is decoded before being displayed... if the bitstream is compromised along the way, it can't be decoded and displayed. So you receive a viewable image or you don't. Decompression does not allow for changing color or grayscale either... you are performing math to restore complete frames from compressed data that describes the frame. There's no room for altering color or grayscale in the decompression process. If the product alters color or grayscale AFTER decompression... that would just be WRONG and it would have to be intentional... not some accidental thing in the product. Or you would have to change a control in the device... like a Blu-ray disc player that has internal color, contrast, brightness, sharpness, etc. settings. If you leave all those on "0" the disc player won't do anything to the decompressed video. But if you move the red control to +10, the disc player will alter the data to make red much stronger. If those controls don't exist in the device, then there's not likely to be any mechanism in the device that would alter color or grayscale.