Can a batsman ask the umpire to check whether the fielder touches the rope?
The players may ask anything they like of the umpires, but few of those requests have bearing in the Laws or can be forced.
The Playing Conditions for all international formats allow a Player Review in the Umpire Decision Review System for whether or not a batsman is out, but not for any other decision. Thus, if there has been no appeal by the fielding team for an out, and therefore no decision by the umpire, a review cannot take place. In this case the umpire must ignore the request for a review and it does not count towards the team's allocation for the innings.
However, it is unnecessary to ask anyway. The bowler's end umpire may ask for the advice of the third umpire if it is not clear whether a boundary has been scored, and the third umpire should advise the umpires if they believe a boundary line "infringement or incident" that has not been acted on. If there would be any reason for the batsman to ask, the umpires would already be checking.
The para 3.1.2 of Appendix D in ICC rules on ICC Men’s Test Match Playing Conditions, effective from 28th September 2017 elaborates on this aspect. It reads "No other decisions made by the umpires are eligible for a Player Review with the exception of Fair Catch/Bump Ball (even after the third umpire has been consulted and the decision communicated)."
Therefore, the batsman can make a request for the review. The outcome of this review can be 1. Out, 2. Not out, 3. Four runs or 4. Six runs.