Why does the instruction “do” require a “while”?

This is largely because creating a separate command would be unnecessary. Adding extra, unneeded commands to a programming language is often considered poor design. Why create a special instruction when one already exists that works?

Your idea of having only one of the "do/while" pair is how it works in other languages, like Python, for instance, since Python doesn't having closing brackets to worry about.

while True:
    ...

It is simply a matter of how any given language is designed. Remember that programming languages are man-made things - they can take whatever form we, as humans, design them to take. You could make your own language that follows the principle you desire, if you felt it was important enough.

Programming languages are still languages, and when translating code from English to programming language X, you may have to change some idioms, reorder words, sentences, or sometimes include unpronounceable curly braces…

When you want an infinite loop, the while (true) or its equivalents can be used. Without good reasons, no language designer will add yet another keyword just for infinite loops.

Your idea that do { ... } always implies a trailing while ... doesn't hold up. In Ruby, do starts a code block, and in Perl it allows you to use a code block on expression level:

# read a whole file at once
my $file_contents = do {
  local $/;
  open my $fh, "<", $filename or die "Can't open $filename: $!";
  <$fh>;
};

Because do is always an expression, various other statement modifers may be used as well (if, unless, and special-cased keywords to evaluate the block once before testing the condition: while and until).

Perl6 is interesting here because it renames the C-style for to the loop keyword. The various statements like initialization are optional, so one can write:

loop {
  say "hi";
}

… which is the same as say "hi" while True.

Just use while (true) – everyone will understand that. The for(;;) is also a widely-understood idiom. If you are using a language with a preprocessor, you could also do evil stuff like #define forever for(;;), but that reduces maintainability.

I would assume that language designers care somewhat about helping developers write accurate code. A language may prefer to not introduce a construct that leads to an infinite loop without the developer explicitly request it. A language like REXX take this concept further and introduces a forever loop as in:

do forever
...
end

So, to answer your question the reasons may be:

1. To help developers write better code.

2. To be similar to other popular languages.

3. To use a single construct instead of several.

I would say that do while is more explicit than just do. In programming it's often beneficial to be explicit. This translates into code that is easier to read. With the proposed do loop the while(true) portion or condition is still implied.

So to sum it all up I'd say it's because most languages are designed to be explicit and readable.

Go has a single instruction for it: for. The for loop has several faces, the simplest form is

for {
    // do something
}

which is equivalent to

for true {
    // do something
}

which is the same as

for ; true ; {
    // do something
}

which is similar to C's for-loop just with first and last part empty.

See the specification.