I hope this could help

http://stackoverflow.com/questions/17781345/calculate-euclidean-distance-of-two-image-in-hsv-color-space-in-matlab

Also, please refer Digital Color Image Processing book by Andreas Koschan, Mongi Abidi, Page 63

https://books.google.ie/books?id=SlXgTyQ86VsC&pg=PA63&lpg=PA63&dq=euclidean+distance+between+two+rgb+colorspace+images&source=bl&ots=5vDLZ9TpEc&sig=enPyhMf2D4-I1AhUZw85cKlMGlE&hl=en&sa=X&ved=0ahUKEwivpdaTydHLAhXEvA8KHeitAAgQ6AEISjAM#v=onepage&q=euclidean%20distance%20between%20two%20rgb%20colorspace%20images&f=false

My colleagues and I  we use the attached here m-file for color and grayscale images.

Best regards 

 V. Gorbachev

Simply treat each color as if they are additional dimensions in your feature space. So your M x N pixels RGB image is just an M x N x 3 array. Transform the array into a vector (using reshape in MATLAB for example). Then use the transformed vectors to calculate the Euclidean distances. 

please make more clear your question

you want compute the euclidean distance between two RGB images for similarity measure or for change detection

for similarity measuring (matching), you will compute the euclidean distance for the whole two images as given by Gorbachev or Till above

for change detection you have to compute pixel wise euclidean distance (the euclidean distance for each same coordinate pixel in the two images) this is given by:

D(x,y)= sqrt((I1G-I2G)^2+(I1B-I2B)^2+(I1R-I2R)^2)

where I1, I2 are RGB images, I1G is the green plan of the I1 RGB image, I1B is the blue plan and I1R the red plan