Pyth, 37 bytes

.U+j;bZ.WgleHJlhH,+_MChZ<eZJ>eZJ,]hQt

Try it online here, or verify all the test cases at once here.

.U+j;bZ.WgleHJlhH,+_MChZ<eZJ>eZJ,]hQtQ Implicit: Q=eval(input())
 Trailing Q inferred
 ]hQ First character of Q, wrapped in an array
 tQ All but the first character of Q
 , 2-element array of the two previous results
 This yields array with rolled carpet (as array of strings) followed by the tail
 .W While condition function is truthy, execute inner function, with initial value of the above:
 gleHJlhH Condition function, input H
 JlhH Number of layers in the current rolled carpet, store in J
 leH Lenth of the tail
 g J Is the above greater than or equal to J?
 ,+_MChZ<eZJ>eZJ Inner function, input Z
 _MChZ Rotate the current rolled carpet (transpose, then reverse each row)
 + <eZJ Append the first J characters of the tail as a new row
 , Pair the above with...
 >eZJ ... all but the first J characters of the tail - this is the new tail
.U+j;bZ Join the carpet roll on newlines and append the tail, implicit print

J, 73 bytes

[:(}:@[,:@:@[,:@:@[,:@:@[,improve this answer

R, 146 132 bytes

function(s)m=F[F]
while(m=rbind(t(m)[,F:0],s[1:F])
s=s[-1:-F]
length(s)>sum(F<-dim(m)))0
write(m[F:1,],1,F[1],,"")
cat(s,sep="")

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Implements the carpet-rolling procedure. Takes input as a list of characters and prints to stdout.

Saved 14 bytes by finding a way to use a do-while loop and initializing using F.

function(s)
m=F[F]					# logical(0); create an empty array (this gets automatically promoted to character(0) later
while(					# do-while loop
 m=rbind(t(m)[,F:0],s[1:F])	# rotate m counterclockwise and add the first F characters of s to the bottom
 s=s[-1:-F]			# remove those characters
 length(s)>sum(F<-dim(m)))0	# while the number of characters remaining is greater than the sum of m's dimensions
write(m[F:1,],1,F[1],,"")		# write the rolled portion write writes down the columns, we reverse each column
cat(s,sep="")				# and write the remaining characters

Jelly, 30 bytes

_{Seems overly long...}

ḢW,ðZU;Ls@¥©ḢWɗ,®Ẏ¤ð/ẈṢƑ$¿ḢY;Ɗ

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How?

ḢW,ðZU;Ls@¥©ḢWɗ,®Ẏ¤ð/ẈṢƑ$¿ḢY;Ɗ - Main Link: list of characters
Ḣ - pop and yield head
 W - wrap in a list
 , - pair with (the remaining list after Ḣ)
 ¿ - while...
 $ - ...condition: last two links as a monad:
 Ẉ - length of each
 Ƒ - is invariant under:
 Ṣ - sort
 / - ...do: reduce by:
 ð ð - the enclosed dyadic chain -- i.e. f(head, tail):
 Z - transpose
 U - reverse each (giving a rotated head)
 ɗ - last three links as a dyad:
 ¥ - last two links as a dyad:
 L - length (i.e. number of rows in current roll)
 @ - with swapped arguments:
 s - split (the tail) into chunks of that length
 © - (copy to register for later)
 Ḣ - pop and yield head (Note register "copy" is altered too)
 W - wrap in a list
 ; - concatenate (the rotated head with the first chunk of the tail)
 ¤ - nilad followed by link(s) as a nilad:
 ® - recall from register (other chunks of tail, or an empty list)
 Ẏ - tighten (the chunks to a flat list)
 , - pair (the concatenate result with the tightened chunks)
 Ɗ - last three links as a monad:
 Ḣ - pop and yield head
 Y - join with newline characters
 ; - concatenate (the remaining tail)
 - when running as a full program implicitly prints