Is it obligatory for someone who took food but did not consume it, and instead gave it away as charity, to pay for its price? If so, is it permissible for him to give a approximate amount as charity instead of paying the restaurant, to avoid embarrassment?

You must pay the owner the value of the food, and it is not sufficient to give charity on his behalf. If you fear scandal, you can deliver the amount indirectly, for example, by sending it through a friend of the rights-holder, explaining the matter to him so he can deliver it. Look for an appropriate way to accurately estimate the value of the food and deliver the right to its owner.