I think he's on both Earths, personally, but I can't recall off the top of my head why I came to that conclusion. We know he's on Earth-2 for sure because Miss Liberty is an ancestress of Liberty Belle, leader of the All-Star Squadron. The case for Tomahawk on Earth-1 is less solid, but based on a JLA-JSA crossover and an appearance in Crisis that both appear to take place on Earth-1, he's probably Earth-1 as well (I use similar reasoning for Black Pirate). There might be some differences between the characters, too. Often for long-running characters from that time period they eventually have stories that directly contradict each other, and in those cases sometimes one story can be placed on Earth-2 and the other story can be placed on Earth-1, even if the majority of the stories could take place on either/both Earths.

According to the DC Fandom website, Tomahawk (the comic) ran from September 1950- June 1972. This means he's one of those "straddler"  characters who exists across the Earth-One/Earth-Two demarcation point in 1955 and who therefore does exist on both Earths: dc.fandom.com/wiki/Tomahawk_Vol_1 . One key difference might be that in the later issues of Tomahawk, his son Hawk was introduced. This occurred in Tomahawk 131 (December 1970), which means that the Earth-One Tomahawk has a son, while his Earth-Two counterpart does not, or has differently named offspring: dc.fandom.com/wiki/Hawk_Haukins_(New_Earth)

Tomahawk also had a group of followers, the Tomahawk Rangers, first introduced in Tomahawk 83 (December 1962). Again, this would seem to indicate that they existed on Earth-One but perhaps not on Earth-Two, unless there are JLA/JSA crossover events that do depict Earth-Two equivalents. What inclusion does exist is ambivalent. Depending on whether the Rangers, Moon Fawn or Hawk show up alongside any Tomahawk appearances in that context, that may mean the character is the Earth-One Tomahawk: dc.fandom.com/wiki/Rangers   
Their final appearance was in Tomahawk 130 (October 1970).

There's also the matter of Moon Fawn, Tomahawk's Apache wife and the mother of Hawk, who also appeared for the first time in Tomahawk 131 (December 1970). It would therefore seem that the Earth-One Tomahawk may have a family, and that his Earth-Two counterpart does not, or may have a differently named wife and offspring. 

The Earth-Two Tomahawk first appeared in Star Spangled Comics 69 (December 1947), with a sidekick called Daniel Hunter. How long that partnership lasted for may be discerned by looking over the Tomahawk comic series precis on the DC Fandom site: dc.fandom.com/wiki/Dan_Hunter_(Earth-Two)   NB: Having done so, I can confirm that this is another 'straddler' character. There seems to be an Earth-One Daniel Hunter as well. Daniel's last appearance alongside Tomahawk occurred in Tomahawk 82 (October 1962), shortly before the arrival of the aforementioned Tomahawk Rangers. If Daniel Hunter shows up alongside Tomahawk in any JLA/JSA crossover/time travel stories, then that may mean the depicted Tomahawk is either from Earth-One or Earth-Two. 

Yep, and Tomahawk met Wonder Girl and Firehawk as well.

Dan Hunter was apparently on different duty with the Revolutionary Army once Tomahawk formed his Rangers. Though he accompanied them on several missions with his last being in Tomahawk Vol 1 #123 July 1968. Dan's appearance in Tomahawk Vol 1 #133 March 1971 was a flashback.
Below is a list of all his appearances

Star-Spangled Comics Vol 1 #69 June 1947 – Star-Spangled Comics Vol 1 #130 July 1952,
Tomahawk Vol 1 #1 September 1950 – Tomahawk Vol 1 #92 May 1964, Tomahawk Vol 1 #95 November 1964, Tomahawk Vol 1 #99 July 1965 - Tomahawk Vol 1 #101 November 1965, Tomahawk Vol 1 #105 July 1966, Tomahawk Vol 1 #114 January 1968 - Tomahawk Vol 1 #116 May 1968, Tomahawk Vol 1 #123 July 1968, Tomahawk Vol 1 #133 March 1971,
World's Finest Vol 1 # 65 July 1953 – World's Finest Vol 1 #88 May 1957, World's Finest Comics Vol 1 #90 September 1957 – World's Finest Comics Vol 1 #101 May 1959.
World's Finest Comics Vol 1 #33 March 1948 – World's Finest Comics Vol 1 #35 July 1948.

I posted a thread here where I listed all the appearances of Tomahawk which mention a specific date. His career is confirmed to span 1750-1795

This also brings up the issue of the E-1 version of Miss Liberty. We know she encountered the Justice League, which places her on their earth, so the only changes to her history should be her descendants. Who knows? There might still be a heroic descendant on E-1 that has yet to make an appearance.