Because a number of movie/TV stars recorded songs offers nothing in the way of proof that Gilligan's Bob Denver did also, but it was interesting seeing the examples.
If he did, I expect that he would not have recorded a 45 on a 39-cent knockoff label like HIT.

Regarding the 2 songs Little Rich linked on the SPAR 45 record credited to a "Bob Denver", Where You Been, composed by Bobby Russell and Pay It No Mind, composed by Bergen White:
Just a sidenote, Bobby Russell composed and sang lead on a rare original release on the HIT label credited to their studio group The Chellos, titled I'm Sure Going To Miss Her, that was covered by Gary Lewis & The Playboys (Sure Gonna Miss Her). 

A YouTube poster also noted about 3 different artist names for Where You Been: 
"If you really want to have your mind blown, this was released 3 different times on 3 different labels. It was released as (under)  "Charlie Brown" on "Country & Western Records," "Charles Baker" on "Hit Records," and, of course, this version (Bob Denver) right here on Spar. The labels were all tied in together and did this pretty often!"

For me, The Smoking Gun: 
Turns out that the Pay It No Mind 45 record is also linked on YouTube to an artist called Tom Bell on Country & Western label, who sounds exactly like the Bob Denver artist on SPAR!  Hmm...the games they played with artist names. In fact there are numerous 45s issued on the Country & Western label under an artist named "Bobby Denver".  Little Rich, perhaps the listener heard one of these?

And I did listen to Where You Been on the SPAR 45, and it sounds to me like Bobby Russell. 
So, I vote no that Gilligan sang either of these 2 songs.

I liked, and still like those old corny beach movies, but I don't recall this one.

I was wondering too if Denver really sang Ho Daddy, Surf's Up. And if it was really him in the sand, how he managed to stay buried for so long during the filming takes.  According to what I "dug up", he did sing it and really was buried.

Due to some snafu, Denver was unaware that he had that singing part, nor had he seen the sheet music until informed on the soundstage that a full orchestra and backup singers were waiting for him to record, which he did, a little at a time. As for the buried in sand scene, it took many takes of lip synching, and the director praised Denver for his patience and for "digging in there, not causing delays".

Source: Hollywood Beach and Surf Movies, The First Wave 1959-1969.  A book by Thomas Lisanti.

Little Rich wrote:

I had one of my call-in listeners put an interestingly wild, wild question to me:
He starts with he heard a song which some alleged was sung by Bob Denver.  A few lines, he the listener, remembered: "South of the California Border" and "Never took the time, never took the time".  He didn't recognize the melody at all and couldn't even sing or hum any of the melody. 
When I researched these clues, I came up with several items from a few of my key web sites.  AllMusic and Wikipedia had absolutely nothing.  However, 45cat.com had an interesting listing including Where You Been composed by Bobby Russell and Pay It No Mind composed by Bergen White.
http://www.45cat.com/record/nc262007us#comments
Both songs back-to-back on one 45 rpm, sung by Bob Denver.  Note in the "Comments", Where You Been has three YouTube links to three people who recorded that song, including Bob Denver.  Additionally, Pay It No Mind has one YouTube link citing only Bob Denver.  Discussions include several people expressing disbelief, or at least skepticism: is it REALLY Bob Denver.  I have my own opinions.  I listened to both songs lyrics and neither mentions any of the aforementioned lyrics my listener provided (South of the California border or Never too the time, never took the time.)
I looked on eBay and found this:
https://www.allmusic.com/album/release/the-spar-records-story-the-singles-collection-mr0003394081
On CD #3, Tracks 13 and 14 list Bob Denver's alleged songs and several songs by Russedll and White are listed.
I found this on YouTube where Denver sings "Ho Daddy, Surf's Up" from a 1964 movie, "For Those Who Think Young."
https://www.google.com/search?client=firefox-b-1-d&q=bob+denver+ho+daddy

What so you on The Board?  Denver a singer or not?!  Thanks
 

When I first saw this post I did a little surfin' and found that Bergen White, the writer of the B side of Denver's single, Pay It No Mind, is still alive and kicking on Facebook. I asked him who sang the song and he finally got back to me today. "Roman, that was actually me singing it. Bobby Russell sang so much of the stuff we did but since I wrote it they wanted me to sing it." Looks like you can lump Gilligan in with Milli Vanilli.