Make a request abstraction is a standard solution but for separation of concerns, your service should be able to make requests but it should return a promise that will be managed by the caller.
Usually the request abstraction manages also 400/401 errors (for refresh tokens/logout) but is agnostic about the logic of the caller.
This is how look like a common abstraction:
 /**
 * Parses the JSON returned by a network request
 *
 * @param  {object} response A response from a network request
 *
 * @return {object}          The parsed JSON from the request
 */
function parseJSON(response) {
  if (response.status === 204 || response.status === 205) {
    return null;
  }
  return response.json();
}
/**
 * Checks if a network request came back fine, and throws an error if not
 *
 * @param  {object} response   A response from a network request
 *
 * @return {object|undefined} Returns either the response, or throws an error
 */
function checkStatus(response) {
  if (response.status >= 200 && response.status < 300) {
    return response;
  }
  const error = new Error(response.statusText);
  error.response = response;
  throw error;
}
/**
 * Requests a URL, returning a promise
 *
 * @param  {string} url       The URL we want to request
 * @param  {object} [options] The options we want to pass to "fetch"
 *
 * @return {object}           The response data
 */
export default function request(url, options) {
  return fetch(url, options)
    .then(checkStatus)
    .then(parseJSON);
}
Then a service will look like this:
import request from 'shared/lib/request';
import { API } from 'shared/constants';
const create = (content) => request(
  {
    url: API.MY_ENDPOINT,
    method: 'POST',
    data: content,
  });
const get = (id) => request(
  {
    url: `${API.MY_ENDPOINT}/${id}`,
    method: 'GET',
  });
const put = (content) => request(
  {
    url: `${API.MY_ENDPOINT}/${content.id}`,
    method: 'PUT',
    data: content,
  });
const MyEndpointService = {
  create,
  get,
  put,
};
export default MyEndpointService;
Usage, wherever you want (also outside react scope):
import MyEndpointService from '/API/MyEndpointService'
MyEndpointService.create(payload)
    .then((data) => {
    // code
    })
    .catch((errors) => {
    // error code
    });